A new algorithm for anisotropic solutions

We establish a new algorithm that generates a new solution to the Einstein field equations, with an anisotropic matter distribution, from a seed isotropic solution. The new solution is expressed in terms of integrals of an isotropic gravitational potential; and the integration can be completed exactly for particular isotropic seed metrics. A good feature of our approach is that the anisotropic solutions necessarily have an isotropic limit. We find two examples of anisotropic solutions which generalise the isothermal sphere and the Schwarzschild interior sphere. Both examples are expressed in closed form involving elementary functions only.     

Intermolecular forces between large molecules: the structure of anthracene stable dimers

Calculations are reported concerning the structure of the anthracene stable dimer. It is shown that the use of an ‘exp-6’ atom-atom potential produces a structure which is definitely not that allowed by the experimental data : this is shown to be due to the absence of any simulation of permanent moment interactions in the atom-atom potentials. A number of conventional methods for calculating the individual terms of the total potential energy are used to estimate the magnitudes of these terms. It is concluded that the resulting relative magnitudes of the individual terms cannot be determined accurately enough to justify the construction of a total potential energy curve. However, by considering the variation of the individual terms with the relative orientation of the two molecules of the dimer, the dominant role of the quadrupole-quadrupole terms in determining the dimer structure is clearly demonstrated.     

A reliable single parameter interatomic potential for argon

A simple semiempirical approximation previously proposed for the isotropic intermolecular forces between two closed shell systems is tested in detail for the argon-argon interaction. The potential is based on the knowledge of the first-order coulomb interaction energy, a suitably damped three term long range asymptotic expansion of the second order coulomb energy, and a semiempirical representation of the exchange interaction energy which contains one adjustable parameter. The single adjustable parameter can be reliably determined by fitting the second virial coefficient for argon in the 130–773 K temperature range with the long range interaction coefficients being constrained within the theoretical bounds specified by Tang, Norbeck and Certain. The reliability of the potential is compared with that of several literature potentials by comparing the theoretical predictions obtained from the potentials with experimental results for the second virial coefficient, viscosity, thermal conductivity and thermal diffusion ratios for dilute argon gas, and with spectroscopic data for the dimer, and with SCF calculations of the Ar-Ar potential at small interatomic separations. Our best potential predicts these properties with a precision as good as or better than other recent potentials which generally contain more adjustable parameters and/or involve more input data. The results confirm earlier work that suggested that the scheme tested is capable of yielding reliable isotropic potentials for the interaction of closed shell systems for 0·3 ? R/R_m ? ∞ where R_m is the intermolecular distance at the van der Waals minimum. The scheme appears to offer a method for obtaining reliable potentials while avoiding problems associated with optimizing many parameters with respect to fitting experimental constraints.     

On the validity of the triple-dipole interaction as a representation of non-additive intermolecular forces

A complete partial wave analysis of the non-expanded non-additive coulomb interaction energy for three non-degenerate S-state atoms is given through third-order in the interatomic potential energy function. Pseudo state techniques are used to evaluate various partial wave components of the non-expanded second and third-order non-additive interaction energies for various isosceles triangular configurations of three interacting ground-state hydrogen atoms. These second and third-order non-expanded coulomb results are used, in conjunction with Heitler-London results for the first-order non-additive energies for the quartet spin state of the H(1s)-H(1s)-H(1s) interaction, to discuss the relative importance of various parts of the non-additive energy as a function of the geometrical configuration of the atoms, and the validity of both the non-expanded triple-dipole energy and the expanded Axilrod-Teller-Muto triple-dipole result as a representation of non-additive coulomb energies. For example, in the non-bonded interaction of three S-state atoms it appears that representing the non-additive energy by the non-additive coulomb energy is not reliable until the interatomic separations are somewhat larger than R*, the interatomic distance associated with the van der Waals minimum in the corresponding non-bonded dimer interaction. Further, the use of the triple-dipole interaction energy, with or without charge overlap corrections, to represent the non-additive coulomb energy is of doubtful validity until the interatomic separations are considerably greater than R*.     

Two-dimensional isotropic scattering in a semi-infinite cylindrical medium

Beginning with the integral equation for the source function, the solutions for the source function, flux and intensity at the boundary of a two-dimensional, isotropically scattering cylindrical medium are found. The incident radiation is collimated and normal to the surface of the medium and depends only on the radial coordinate. For a Bessel function boundary condition, separation of variables is used to reduce the source function integral equation to a one-dimensional equation. The resulting integral equation is shown to be the same as that for the two-dimensional planar case. Solutions for other boundary conditions are then shown to be superpositions of the Bessel function solution. Numerical results are presented for a Gaussian distribution of incident radiation which closely models a laser beam. These multiple scattering results are compared to the single scattering approximation. Also, the solution for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function is expressed in terms of the isotropic results.     

Anisotropic intermolecular potential from nuclear spin-lattice relaxation in hexafluoride gases

The data on fluorine spin-lattice relaxation times per unit densityT _jσ in pure SF₆ and UF₆ gases can be analyzed to obtain information on the anisotropic part of the intermolecular potential in these systems. A new and more performant potential, Morse-Morse-Spline-van der Waals potential (J. Chem. Phys.94, 1034 (1991)) was used for the isotropic part of the intermolecular interaction. The analysis was made using the Bloom-Oppenheim theory, assuming, that the correlation time of the spin-rotation interaction can be approximated by the average lifetime of a molecule in a givenJ state. We have obtained the strengths of the repulsive and attractive terms in the anisotropic potential. From the strength of the attractive term, the hexadecapole moment of SF6 and UF6 were also obtained, being in good agreement with the values reported earlier, based on other potentials and techniques.     

On the validity and properties of the atom-atom potential as a function of intermolecular separation,configuration, and partial wave order

The intermolecular partial wave expansion of the atom-atom potential U is reviewed briefly and developed, by using results due to Sack, so that the radial components of the expansion can be evaluated to arbitrary accuracy for all relevant partial wave orders and values of the intermolecular distance r. These results are used to study the convergence of the partial wave expansion of U as a function of partial wave order, r, intermolecular orientation, and the anisotropy of the interacting molecules. In marked contrast to previous work it is found that many of the higher order partial wave components of U are important relative to the isotropic term even for the interaction of relatively spherical molecules and that the results obtained from a truncated partial wave expansion depend significantly upon the method of summation due to the generally poor convergence of the expansion. The validity of the atom-atom potential as a representation of the correct attractive intermolecular potential is also discussed in some detail. There are basic problems associated with the representations furnished by both the isotropic and the anisotropic parts of the atom-atom potential at intermediate and large r. The different convergence properties of the r ^-1 expansions of the partial wave expansions of U and of the correct potential for these values of r is illustrated by using model interactions. While it appears that it may be possible to obtain a qualitatively reasonable representation of the attractive part of an intermolecular potential over a useful range of r from atom-atom results, this apparently cannot be achieved for wider ranges of r or for the purely anisotropic part of the potential.     

Long range C,N and H atom-atom potential parameters from ab initio dispersion energies for different azabenzene dimers

After comparing theoretically the atom-atom model with the multipole expansion for the dispersion energy, we derive atom-atom potential parameters for C, N and H contacts by fitting the atom-atom dispersion energies to the dispersion interactions calculated by an ab initio method for the following (aza)-benzene molecules: benzene, pyridine, pyridazine, pyrimidine, pyrazine, s-triazine and s-tetrazine. The data base for the fit consists of 1200 randomly chosen configurations for each azabenzene dimer. The optimized parameters for the carbon and hydrogen contacts in particular are not unique ; reasonably good fits are obtained with different parameter sets. A very simple rule, which relates the atom-atom potentials to the isotropic molecular C₆ dispersion coefficients, already leads to good estimates for the parameters. From the empirical parameter sets available in the literature the William-Govers set yields results which are close to our ab initio dispersion energies.     

Perturbation theory for equilibrium properties of molecular fluids

Perturbation theory for the angular pair correlation function g(r₁₂ ω ₁ ω ₂), using a fluid with isotropic intermolecular forces as the reference system, is applied to the calculation of a variety of macroscopic properties. Comparisons with experiment are made for methane, oxygen and nitrogen (and carbon monoxide for infra-red and Raman band moments) in the dense fluid and liquid states. Theoretical expressions are given and calculations made for thermodynamic properties (isothermal compressibility, pressure, configurational energy, entropy and specific heat) both along and away from the vapour-liquid co-existence curve, for infra-red and Raman band moments, and for neutron scattering cross sections. Excellent agreement with experiment is obtained for all properties, except for the infra-red and Raman band moments; this latter comparison is inconclusive because of large experimental uncertainties. The anisotropic intermolecular forces are found to have very little effect on the liquid isothermal compressibility, in agreement with the first-order theory. Molecular anisotropy has a relatively small effect on the configurational energy and on the Helmholtz free energy, but the effect is large for pressure and specific heat. The pressure is more sensitive to short-range anisotropic forces than the other properties, whereas the specific heat is particularly sensitive to the long-range anisotropic forces. Mean squared torques (derived from infra-red and Raman band moments) are very sensitive to the strengths of the anisotropic forces, and are more sensitive to higher terms in the multipole series than are the other properties. The structure factors for oxygen and nitrogen are found to be little affected by the anisotropic forces.     

Local structure and triplet energy migration in p-dichlorobenzene-p-dibromobenzene solid solutions

Studies of the first triplet absorption and emission spectra of p-dichlorobenzene-p-dibromobenzene (DCB-DBB) mixed crystals agree generally with the data, indicating that this system forms solid solutions. However, the spectral properties are found to be very different from that of isotopically mixed crystals of comparable concentrations. The singlet-triplet absorption spectra of the DCB-DBB mixtures are generally broad and not amenable to detailed analysis indicating significant site dependent perturbations of the DCB and DBB triplet energies. Energy migration enhances emission from the lower energy sites and in consequence a red shift in the position of the emission (0,0) is observed, which is a maximum at equimolar concentrations. The emission bands are generally much broader than found for isotopically mixed crystals, indicating that the many types of sites in the DCB-DBB system remained uncoupled. This is a direct consequence of exciton trapping by inhomogeneous energy broadening caused by the site energy disorder being greater than the triplet exciton band width. Long-range triplet energy migration is not observed at DCB concentrations less than 99 per cent. This high limit is only expected for near one-dimension energy migration topologies. New structure observed at low DCB concentrations is tentatively interpreted as being due to the formation of DCB n-mers. The observed splittings indicate that the gas-to-crystal shift of the DCB T ₁ state increases (becomes more negative) by 11 cm^-1 when a neighbouring DBB molecule is replaced by DCB.     

On the theory of conductivity of anisotropic composites: A linear-concentration approximation of inclusions

A general approach is proposed for calculating the conductivity of anisotropic composites with a low concentration of inclusions of an arbitrary shape. The contribution to effective conductivity [^(s)] _e\hat \sigma _e, which is linear in concentration, is expressed in terms of the polarizability of the inclusion defined in a certain transformed system in which the inclusion is surrounded by an isotropic matrix. A transition to this system is performed using a symmetry transformation that does not change the equations for direct current.     

Temperature response in participating media with anisotropic scattering caused by pulsed lasers

It is not by isotropic scattering but by anisotropic scattering that radiant energy is redistributed in some materials containing real particles, fibers, or impurities. In some instances, great difference can be caused in transient thermal behavior between isotropic scattering and anisotropic scattering media. Ray tracing method combined with Hottel's zonal method is introduced to deduce thermal radiative source term for various optical boundary conditions induced by collimated incidence passing through translucent boundary. Temperature response caused by laser pulse at non-incident side of participating and anisotropic scattering media is examined. We investigate effects of scattering albedo, scattering phase function, initial temperature of media and thickness of media on temperature response. Results obtained for anisotropic scattering media are compared with those for isotropic scattering one and show that anisotropic scattering must be considered in the simulating measurement of thermophysical properties by the laser flash method for some materials with big scattering albedo which behave anisotropically, or big error will be introduced; forward scattering can increase excess temperature and backward scattering can decrease it at non-incident side of the considered sample irradiated by laser pulse.     

Rigid rod adsorption including long-range dispersion interactions

The Caillé-Ågren analysis of rigid rod adsorption is extended by applying the van Kampen theory of condensation to include long-range dispersion interactions between the adsorbed rods. We discuss in greater detail the characteristics of the phase transition predicted by Caillé and Ågren between states having an isotropic and anisotropic distribution of rods adsorbed parallel to the surface. The maximum density range over which the anisotropic adsorbed phase is stable is determined as a function of the length-to-breadth ratio x of the rigid rods and the strength of the anisotropic dispersion energy. Critical surface adhesion and anisotropic dispersion energies necessary for anisotropic adsorption are also obtained as a function of x. In agreement with Caillé and Ågren the isotropic-anisotropic transition for rigid rod adsorption with attractive forces present is found to be second order. We also discuss the spreading pressure-density or area per molecule isotherms obtained for adsorbed rods having various values of x and surface adhesion and dispersion energies. Whenever feasible we compare our results with the spreading pressure isotherms obtained for monolayers of lyotropic molecules on either aqueous or mercury subphases and obtain qualitative agreement. In particular, the critical density and pressure associated with the two dimensional adsorbed gas-liquid condensation for rigid rods with x = 10 gives reasonable agreement with the critical constants observed by Hawkins and Benedek and Kim and Cannell for the corresponding condensation of pentadecanoic acid monolayers on neutral and acidified aqueous subphases. This agreement suggests that considerable dimerization of the pentadecanoic acid molecules may occur on aqueous surfaces.     

Emissivity estimates

A variational expression is developed to estimate the overall and angular emissivities for a halfspace geometry described by the equation of transfer. In contrast to the usual variational procedure of postulating a functional and then proving correctness, the appropriate functional is derived using a Lagrange multiplier technique. With asymptotic trial functions, simple expressions for the emissivities result. This variational estimate of the overall emissivity is exceeding accurate when compared to exact results for an isotropic phase function. The angular emissivity, a more detailed quantity, is estimated with less accuracy. The formalism is not restricted to an isotropic phase function, but is valid for general anisotropic scattering. As an example of an anisotropic phase function, the case of Thomson scattering is considered. The emissivity for this phase function is found to be slightly larger than that for isotropic scattering.     

Influence of cubic anisotropy in dipole-dipole interactions on structural phase transitions using the renormalization group transformation

The critical behaviour of ferroelectrics is studied forT>T _c and for zero electric field, by the renormalization group techniques ind=3 dimensions. The model hamiltonian we used contains an isotropic exchange coupling and a dipolar interaction with strong cubic anisotropy. A quadratic anisotropic fixed point is found to be stable for the cubic anisotropic scale field. Experiments on BaTiO₃ and KTaO₃ are compared with our results.Work supported by N.F.W.O.     

Elastic interaction between defects in thin and 2D films

Elastic interactions between defects is investigated at the surface of thin layers, a question on which we have given a brief account [P. Peyla et al. Phys. Rev. Lett. 82, 787 (1999)]. Two isotropic defects do not interact in an unlimited medium, regardless of the spatial dimension, a result which can be shown on the basis of the Gauss theorem in electrostatics. Within isotropic elasticity theory, defects interact only (i) if they are, for example, at a surface (or at least if they feel a boundary), or if their action on the material is anisotropic (e.g. they create a non central force distribution, though the material elasticity is isotropic). It is known that two identical isotropic defects on the surface of a semi-infinite material repel each other. The repulsion law behaves as 1/r ³(r = defects separation). We first revisit the Lau-Kohn theory and extend it to anisotropic defects. Anisotropy is found to lead to attraction. We show that in thin films defects may either attract or repel each other depending on the local geometric force distribution caused by the defect. It is shown that the force distribution (or more precisely the forces configuration symmetry) fixes the exponent in the power law 1/r ⁿ (e.g. for a four-fold symmetry n = 4). We discuss the implication of this behaviour in various situations. We treat the interactions in terms of the symmetries associated with the defect. We argue that if the defects are isotropic, then their effective interaction in an unlimited 2D (or a thin film) medium arises from the induced interaction, which behaves as 1/r ⁴ for any defect symmetry. We shall also comment on the contribution to the interaction which arises from flexion of thin films. Received 7 October 2002 Published online 4 June 2003 RID="a" ID="a"e-mail: chaouqi.misbah@ujf-grenoble.fr     

Elastic interactions of adatoms or clusters on a surface: Effects due to anisotropy of the substrate and finite size of the clusters

The interactions of atoms or clusters of atoms adsorbed on a surface due to elastic strains created in the substrate is investigated. At large distances it falls off ass ^–3. The strength of the interaction which is repulsive for elastically isotropic substrates becomes angular dependent in general and becomes even attractive in certain directions for substrates with sufficient anisotropy. For a certain model of a cluster the interaction among clusters is calculated at all distances for the isotropic as well as for the anisotropic case. A repulsive potential barrier is found at distances of the order of the cluster diameter even in situations where the long-range behaviour is attractive.     

Collision-induced light scattering spectra of mercury vapour at different temperatures

The two-body isotropic and anisotropic collision-induced light scattering spectra of mercury vapour at different temperatures are analysed in terms of new interatomic potential and interaction-induced pair-polarisability trace and anisotropy models, using quantum line-shape computations. The quality of the present potential has been checked by comparing between calculated and experimental spectroscopic, thermo-physical and transport properties over a wide temperature range, which are found to be in good agreement. Also, the comparison of the computed spectra with a measurement at different temperatures permits improvements on the existing trace and anisotropy models of the pair polarisability.     

Abdominal apparent diffusion coefficient measurements: effect of diffusion-weighted image quality and usefulness of anisotropic images

This study aimed to assess the effect of diffusion-weighted image (DWI) quality on abdominal apparent diffusion coefficient (ADC) measurements and the usefulness of anisotropic images. Twenty-six patients (10 men and 16 women; mean, 58.1 years) who underwent DW imaging and were diagnosed not to have any abdominal diseases were analyzed. Single-shot spin-echo echo-planar DW imaging was performed, and one isotropic and three orthogonal anisotropic images were created. ADCs were calculated for liver (four segments), spleen, pancreas (head, body, tail) and renal parenchyma. Image quality for each organ part was scored visually. We estimated the correlation between ADC and image quality and evaluated the feasibility of using anisotropic images. ADCs and image quality were affected by motion probing gradient directions in the liver and pancreas. A significant inverse correlation was found between ADC and image quality. The r values for isotropic images were −.46, −.48, −.70 and −.28 for the liver, spleen, pancreas and renal parenchyma, respectively. Anisotropic images had the best quality and lowest ADC in at least one organ part in 17 patients. DWIs with the best quality among isotropic and anisotropic images should be used in the liver and pancreas.     

A new universality for random sequential deposition of needles

Percolation and jamming phenomena are investigated for random sequential deposition of rectangular needles on d=2 square lattices. Associated thresholds and are determined for various needle sizes. Their ratios are found to be a constant for all sizes. In addition the ratio of jamming thresholds for respectively square blocks and needles is also found to be a constant . These constants exhibit some universal connexion in the geometry of jamming and percolation for both anisotropic shapes (needles versus square lattices) and isotropic shapes (square blocks on square lattices). A universal empirical law is proposed for all three thresholds as a function of a. Received 27 January 2000 and Received in final form 2 February 2000