Exact Numerical Data Research Articles

Caliper diameter of branched polymers

The authors report analyses of exact numerical data for the spanning diameter of two-dimensional lattice animals up to size N=17. Estimates of the exponent nu are consistent with previous studies. The leading correction to scaling has an exponent sigma approximately= nu which does not result from irrelevant variable effects. Interpretation of this correction as a 'surface' term is proposed.

Spectral-density method for classical systems: Heisenberg ferromagnet

We formulate the spectral-density method in classical statistical mechanics in strict analogy with the known quantum version, and we apply it to the classical Heisenberg ferromagnetic model in an external field. A new formula for the magnetization in the classical formulation of spin-vector Green functions is derived for arbitrary spatial dimensionality. Furthermore, the static properties of the one- and three-dimensional cases are considered in detail. We obtain accurate results over a wide range of temperatures with the external field for the one-dimensional model and without field for the three-dimensional model. In the first case very good agreement with the exact numerical transfer-matrix data is also found in the region of higher temperature where the interacting-boson approach fails. In particular the zero-field susceptibility results are found to be consistent with the exact results obtained by Fisher.

Large-order expansion of the imaginary part of the Stark effect in hydrogen and one of the anharmonic oscillator with negative anharmonicity

The asymptotic formula in N for the coefficients of expansion in the field strength for the Stark level width in hydrogen and a similar formula for the two-dimensional anharmonic oscillator with negative anharmonicity are derived. A comparison with exact numerical data is given.

On ionization “threshold” for excited atomic states in a uniform electric field

The exact numerical data for the ionization thresholds of hydrogen atom for n = 5 states are presented. The applicability of the approximate methods used to estimate “thresholds” is discussed.

Self-avoiding walks on irregular networks

Exact numerical data on self-avoiding walks are presented for the irregular network studied by Finney (1970) in connection with the Bernal model of a liquid and for a face-centred cubic lattice with randomly removed bonds. Averages for the total number of walks Cn, and polygonal closures Un, are defined and found to be consistent with Cn approximately mu nng, Un approximately mu nnh where the critical parameters g and h have the same values as those obtained from the regular lattices. The mean-square end-to-end distances (Rn2) are also studied and the critical parameter theta is found to have the usually accepted value of 6/5. The results add further support to the conjecture that these exponents depend only on the dimensionality and are not affected by the irregularity of the network.

Approximate analysis of a multicomponent reacting wake

An integral method is proposed for analysis of a multicomponent reacting far wake behind a body. The results of the analysis, obtained by using the method proposed, are compared with exact numerical data.

Electron scattering cross sections with relativistic correction in the first Born approximation for Thomas-Fermi and Hartree potentials

In this paper we calculate the electron scattering cross sections which include the effect due to relativity in the first Born apptoximation for Thomas-Fermi and Hartree potentials The Thomas-Fermi potential and the Hartree potentials are approximated by analytica fits. The results of Table I show that the electron scattering cross sections without relativistic correction agree quite well with the numerical results of the exact Thomas-Fermi potential. In Table II we have a comparison of electron scattering cross sections for approximate Thomas-Fermi and approximate Hartree potentials with the exact numerical data calculated from Hartree fields for argon. Table II shows that the analytical fits for Hartree fields do not differ essentially from the fits for Thomas-Fermi potential. In Table III we have collected 19 fits of Hartree potentials for neutral atoms. We derive an analytical approximate formula for the electron scattering total cross section which includes relativistic correction. In Table IV we have several numerical data for the electron scattering total cross section without the relativistic correction for argon for the Thomas-Fermi and Hartree potential.

On the capillary forces in an ideal soil; correction of formulae given by W. B. Haines

1. The omission of the tension in the air-water interface has introduced an erroneous factor into Haines’ formulae; certain additional factors have also crept into his expressions for average stress.2. With these corrections, the stress due to moisture varies comparatively little with changing water content, though falling slightly throughout the range. The energy needed to cause rupture rises continuously in a manner not unlike Haines’ measurements, and should more probably be associated with them than should the tensile stress.3. The geometrical approximation used by Haines gives a close geometrical representation of the figure, but a mechanical approximation which is less satisfactory. Since neither the formulae connected with the true curve, nor the tables needed to use them, are readily, accessible, sufficiently exact numerical data have been here put on record.

3. Theoretical Investigations into the same.

Having brought before this Society in April 1849, a plan for cooling the air of rooms in tropical climates, the author was anxious to determine by actual experiment on a very large scale the practicability of the principle involved, viz., the thermotic effect of the compression of air. He had had a small apparatus made in 1844, which, though not sufficiently large to give exact numerical data, at least showed that the plan was in the bounds of possibility.