Reciprocal Length Research Articles

A theoretical model of neutron thermalization in a medium

A simple theoretical model is developed to explain neutron thermalization properties. Assuming that a fast neutron source is located at the origin, the neutron distribution is described as a function of distance from the source. Based on the continuous slowing-down model, it is shown that the thermal neutron flux is a simple function of the diffusion constant D, the reciprocal length κ, and the Fermi age τ of a moderator material. Several points are noteworthy from calculation of the thermal neutron flux. First, theoretical results for polyethylene moderator agree remarkably well with simulation data obtained from a Monte Carlo simulation. Second, the thermal neutron flux at the origin is proportional to the square of moderator density. Third, the volume of the large thermal neutron flux increases drastically as the Fermi age increases. Finally, we also note that theoretical results agree reasonably well with experimental data for a broad range of physical parameters. A simple analytical expression of the thermalization factor of a moderator material is derived and it is found that its theoretical values agree rather closely with measured values.

Low‐frequency raman‐active modes in methoxy‐ended and hydroxy‐ended oligo(oxyethylene)s

AbstractLow‐frequency Raman spectra were recorded for α‐methyl, ω‐methoxy‐oligo(oxyethylene)s, C1EmC1 with m in the range 4–25 (i.e., 15–78 atoms). Longitudinal acoustical mode (LAM‐1) frequencies were identified and compared with those determined previously for α‐hydro, ω‐hydroxy‐oligo (oxyethylene)s. Nonlinear relationships between LAM‐1 frequency and reciprocal chain length were explained as an effect of intermolecular end forces.

A computational study on secondary minimum depth

Extensive computation has been carried out on particle interaction potential energy secondary minimum depth. The minimum depth is directly proportional to the effective particle radius. Almost linear log—log relations have been found between the secondary minimum depth and double-layer potentials, Hamaker constants and Debye reciprocal length. A simple approximate formula is proposed for the rough evaluation of the minimum depth. According to the almost linear log—log relations, secondary minimum depth of an unequal Hamaker constant interaction or an unequal potential interaction is the geometrical mean of the minimum depths of the corresponding two dual Hamaker constant interactions or equal potential interactions.

Crystallinity of uniform oligo(oxyethylene) mono-alkyl ethers studied by Raman spectroscopy

Low-frequency Raman spectra have been recorded for two series of uniform oligo(oxyethylene) mono-n-alkyl ethers: α-alkyl,ω-hydroxyoligo(oxyethylene)s, H(CH2)n(OCH2CH2)mOH, and α-alkyl,ω-methoxyoligo(oxyethylene)s, H(CH2)n(OCH2CH2)mOCH3. LAM-1 frequencies were identified and compared with those determined previously for n-alkanes, oligo(oxyethylene) di-n-alkyl ethers and oligo(oxyethylene) dimethyl ethers. On the basis of the linear crystal model of Minoni and Zerbi (J. Phys. Chem., 1982, 86, 4791), non-linear relationships between whole-chain LAM-1 frequency and reciprocal chain length have been explained as effects of end masses and end forces. Prominent bands in the spectra of the α-alkyl,ω-hydroxyoligo(oxyethylene)s were assigned to LAM-1 and LAM-3 of the H-bonded dimer crystallised in a bilayer structure.

On the theory of gel electrophoresis of DNA: Extension and evaluation of the Lumpkin-DèJardin-Zimm model

Using fundamental concepts of hydrodynamics in porous media, we have rederived the Lumpkin-DèJardin-Zimm (LDZ) model for the gel electrophoresis of reptating, infinitely long, worm-like chains, such as DNA. The force balance provides a constraint for evaluating the correlation among the segment-to-field angles of a given molecular conformation. We have used an approximate analytical expression to account for this correlation in order to apply the present derivation to finite chain lengths. The resulting extended LDZ model predicts a nonlinear variation of electrophoretic mobility (μ) with reciprocal chain length ( 1 L c ) at low electric field strengths, similar to the one observed. The present derivation is valid only at low electric field strengths, and the predictions of the extended LDZ model fit data for a dimensionless electric field strength, E 1 ∗ , of less than 1.23. An empirically useful criterion for determining the onset of reptation is also described. The present treatment shows how size-exclusion effects can be included in future theories. Models based on reptation alone are shown to predict a discontinuity in the molecular chain length dependence of mobility at a critical molecular size. Such discontinuities are not observed experimentally.

Effects of surface roughness (hairiness) of latex particles on their electrokinetic potentials

In a previous paper, we demonstrated that the electric properties of the bitumen/water and crude oil/water interfaces could be explained by the dissociation of carboxyl groups belonging to natural surfactants present in bitumen. Electrophoresis of bitumen-in-water emulsions indicated very high zeta potentials (−100 mV in 10 −2 M NaCl) especially when compared to a carboxylated latex (−50 mV in 10 −2 M NaCl) having similar surface charge density. This difference was attributed to the surface roughness of the latex particle, which results in a larger value for the location of the shear plane resulting in a smaller value for the zeta potential. To test this hypothesis we modified the surface roughness of the latex by heating it above its glass transition temperature. The heat treatment resulted in a significant increase of the zeta potentials calculated from the measured electrophoretic mobilities. This observation suggests that surface roughness is one of the most important factors in determining zeta potentials from electrophoresis. Results of electrophoretic mobilities of the original and heat-treated latexes will be shown as a function of pH in a wide range of NaCl concentrations, and will be compared with those predicted by the Ionizable Surface-Group model. In addition, results of measured mobilities of the heat-treated latex in NaCl, CaCl 2, and La(NO 3) 3 below 10 −4 M suggest that the computer solution of O'Brien and White overestimates the electrokinetic relaxation effect of itκa < 100, where κ is the reciprocal Debye-Huckel length and a is the particle radius.

Site-binding model with long-range co-operative interaction for a polyion with large and small ionic ligands in competition

This model is related to the regulative influence of ionic strength on protein—nucleic acid interaction. The interaction of ions and polyions has hitherto been described in terms of ion condensation. An alternative approach is a site-binding model. By means of statistical thermodynamics and electrolyte theory, a model for competitive site binding of large and small ionic ligands at a polyion was developed. This model was treated by both the method of sequence generating functions and the method of conditional probabilities in coupled binding equilibria. Common usage of both these methods is helpful in finding a suitable description of long-range interaction by long-range factors modifying nearest-neighbour interaction parameters. Ionic contributions to the interaction energy were calculated by modified Debye—Hückel theory as functions of charge distance parameters and ionic strength I via k, the Debye—Hückel reciprocal length. By means of computerized simulation we studied: 1. the contribution of different types of interaction to the binding behaviour as a function of the ionic strength; and 2. the influence of ligand charge modification on the binding behaviour.

Site-binding model with long-range co-operative interaction for a polyion with large and small ionic ligands in competition

This model is related to the regulative influence of ionic strength on protein-nucleic acid interaction. The interaction of ions and polyions has hitherto been described in terms of ion condensation. An alternative approach is a site-binding model. By means of statistical thermodynamics and electrolyte theory, a model for competitive site binding of large and small ionic hgands at a polyion was developed. This model was treated by both the method of sequence generating functions and the method of conditional probabilities in coupled binding equilibria. Common usage of both these methods is helpful in finding a suitable description of long-range interaction by long-range factors modifying nearestneighbour interaction parameters. Ionic contributions to the interaction energy were calculated by modified Debye-Hückel theory as functions of charge distance parameters and ionic strength I via κ, the Debye-Hückel reciprocal length. By means of computerized simulation we studied: • the contribution of different types of interaction to the binding behaviour as a function of the ionic strength; and • the influence of ligand charge modification on the binding behaviour.

An approximation of the second dissociation constant for H 2S

Poor resolution ΔG ƒ 0 for S 2− affects hydrothermal complexing arguments, evaluation of sulfide solubility products, and calculation of aqueous sulfur speciation. The second dissociation constant for H 2S ( log K HS− ) if well known could be used to determine this free-energy. The log K HS− has been evaluated here by extrapolating the thermodynamic data for the dissociation of polysulfides (H 2S n) as a function of the reciprocal chain length ( 1 n ). The extrapolation method is supported by standard weak acid theory. The extrapolation is further based on flocculation experiments which determined the equilibrium constant for the protonation of a colloidal sulfur surface, which is a reaction analog for the protonation of an infinite sulfur chain, H 2S∞. The derived log K HS− value, −18.51 ± 0.56 (20°C), implies that S 2− is never a dominant aqueous species. This new value has been used to derive alternative solubility products of some metal sulfides.

Activation gaps in the energy spectrum and influence of disorder under fractional quantum Hall effect in silicon MOSFET

In Si MOS structures with high mobility of 2D electrons in a strong magnetic field (H ≲ 20) applied perpendicularly to the space charge layer, the activated conductivity is used for determination of activation gaps W in the energy spectrum of interacting electrons at the fractional filling factors: ν = 13, 23, 43and45. It was found that these gaps increase proportionally to the reciprocal magnetic length under the condition that the electron mobility μe is fixed with an accuracy of about ±3%. The dependence of the activation gaps on H and μe can be factorized within the simple relation: W ∼ √H(1 - const./μe. The magnitudes found for W, and their dependence on H and μe, correlate with the theoretical conception concerning the condensation of 2D electrons into the quantum incompressible Fermi fluid.

Activation gaps in the energy spectrum and influence of disorder under fractional quantum Hall effect in silicon MOSFET

In Si MOS structures with high mobility of 2D electrons in a strong magnetic field ( H ≲ 20) applied perpendicularly to the space charge layer, the activated conductivity is used for determination of activation gaps W in the energy spectrum of interacting electrons at the fractional filling factors: ν = 1 3 , 2 3 , 4 3 and 4 5 . It was found that these gaps increase proportionally to the reciprocal magnetic length under the condition that the electron mobility μ e is fixed with an accuracy of about ±3%. The dependence of the activation gaps on H and μ e can be factorized within the simple relation: W ∼ √ H(1 - const./ μ e. The magnitudes found for W, and their dependence on H and μ e, correlate with the theoretical conception concerning the condensation of 2D electrons into the quantum incompressible Fermi fluid.

Sunspot Number Series Envelope and Phase

The sunspot number series R( t) from 1700 to date is found to be representable by R( t) = I Jf' {Re( E( t) exp[i {wo t + ( t) I]) + U( t) 1 I, where Wo is the angular frequency corresponding to a period of 22 years, E(t) is the instantaneous envelope amplitude, (t) is the instantaneous phase of a complex time-varying analytic function, U(t) is an undulation of low amplitude and period about 30 (22-year) cycles and jy is a nonlinear operator whose main effect is to introduce a small amount of third harmonic (period about 7 years). The justification for the 22-year period is the known fact that the observable sunspot magnetic fields reverse polarity every 11 years or so at the time of sunspot minimum; the undulation has been demonstrated, and its period determined, in fossil records discovered by Williams; and the third harmonic is an expected consequence of minor nonlinearity in the dependence of the arbitrarily defined R( t) on the physical cause of sunspots. The algebraic representation is established by the Hilbert transform method of forming a complex analytic function as proposed by Gabor. The method reveals three obscuring features that may be alleviated as follows: use the alternating series R� (t) in which alternate II-year cycles take opposite signs, remove the third harmonic, and subtract the undulation. These justifiable steps remove artificial components, such as sum and difference frequencies, that are gratuitously and nonlinearly introduced by conventional Fourier analysis as applied to the rectified, or absolute, value of the 22-year oscillation. Then a complex envelope E( t) exp {i ( t)j can be discerned whose intrinsic behaviour can be studied to reveal statistics that bear on the physical origin of the solar cycle. The results favour a deep monochromatic oscillator whose influence is propagated to the observable surface via a time-varying medium. The r.m.s. value of the component of E(t) is 0�4 of the mean and the characteristic time is a century. Frequency analysis of the envelope does not support a 78-year period in the modulation noticed by Wolf. Both the statistical frequency distribution of the amplitude E( t) and its spectrum are subject to refinement by analysis of fossil solar records. The results do not favour the theory that the 22-year period is set by the natural frequency of a resonator with characteristic damping subject to random turbulent excitation. Also disfavoured is the theory of energy release at intervals determined by a relaxation process. Correlation has been found between the phase departure ~(t) from linear and envelope amplitude and attributed to propagation of the magnetic .cycles through a time-varying, such as a convecting, medium. A correlation not depending on Hilbert transform analysis is predicted between the reciprocal cycle length and envelope amplitude and found to� exist. Variability of the sunspot cycle length can be viewed as a Doppler shift due to propagation in a time-varying medium and the Wolf modulation then represents the concomitant intensity change. Agreement has been found between E(t) and '(t) but not explained. If the explanation is dispersion in the propagation of the assumed magnetic flux waves then there is a mode of oscillation. that has the characteristics required for the undulation U( t). Extra buoyancy possessed by the magnetic field of strong cycles accounts for the fast rise time of strong cycles.

Variation of the potential of the stern plane with interaction of dissimilar flat electrical double layers

The potential, Ψ δ , of the Stern planes of dissimilar double layers were calculated as a function of the separation distance. Also the surface charges, the Stern layer charges, and the diffuse layer charges were computed under the assumption of constant surface potentials. The dependence of Ψ δ on the parameter κ d (κ is the Debye reciprocal length parameter, d is half distance of separation) is distinct when κd < 1, particularly when the specific adsorption potential, φ, ∼0. As φ increases to ∼5 kT the dependence of Ψ δ on κ d is less distinct. It was found that the linear approximation is allowed for the system of high specific adsorption potential and high value of κ.

On the inhomogeneous two-plasmon instability

The two-plasmon instability in warm inhomogeneous plasma for a normally incident pump is considered. The complex eigenfrequencies of the absolute instability are obtained by reducing the linearized fluid equations to a Schrödinger equation in wavenumber space. These eigenvalues are obtained in several ways. One is by combining a perturbation expansion in powers of the reciprocal scale length with WKB theory. The resulting algebraic equations are solved by three analytical approximations and by direct numerical solution. A second way is by analysis of the Schrödinger equation using an interactive WKB computer code. A third way is by the use of a shooting code. These methods are all used and compared for threshold curves and growth rates above threshold. Some eigenfunction forms are also obtained. The threshold is near (v0/ve)2k0 L =3, and varies weakly with β≂v4e/v20c2, rising from near 2 to about 4 over six decades of variation of β. The corresponding critical value of (ky/k0)2 is near 0.2/β over this range. Above threshold, there is a smooth variation of the growth rate with (ky/k0)2, peaking at some intermediate value. The perturbation method is in good agreement there with more exact calculations. Experimental implications of these results are discussed.

A two-parameter perturbation series for the reciprocal length of polymer chains and subchains

Generating functions are defined which permit the computation of moments of the end-to-end length of random walks. These functions are applied, within the framework of the Domb-Joyce (1972) model, to the development of a two-parameter perturbation series for the contraction factor alpha ij-1 of the mean reciprocal distance (Rij-1) between the segments i and j of self-avoiding walks. The case where i or j denotes a chain end is shown to be fundamentally different from the case where neither denotes a chain end.

Electrostatic potential between concentric surfaces: Spherical, cylindrical, and planar

An approximate closed-form solution to the one-dimensional Poisson-Boltzmann equation is obtained for any combination of mono-, di-, and trivalent counterions. This solution provides the electrostatic potential between two apposed concentric surfaces, i.e., flat plates or a large sphere or cylinder embedded in a cavity of the same geometry. The solutions for cylindrical and spherical surfaces are quite accurate provided that their radius ( a 1) is much larger than the double layer thickness; specifically, that κa 1 > 10 where κ is the Debye-Hückel reciprocal length. The spherical or cylindrical solutions apply to the cell model of particle aggregates, wherein the electrostatic environment of an average particle within an aggregate may be estimated. Simple approximate formulas for the ion distributions in the double layer are also obtained.

A theoretical approach to the magnetic flocculation of weakly magnetic minerals

Energy of interaction of a system of mineral paramagnetic and diamagnetic particles of colloidal dimensions is calculated under the influence of external magnetic field according to DLVO theory. Conditions of rapid flocculation into a potential well are derived in terms of threshold magnetic field as a function of particle size and magnetic susceptibility. The effect of surface potential, Debye-Hückel reciprocal length parameter and Hamaker constant is briefly discussed. The values of the flocculating magnetic field correspond more closely to reality than other theoretical models.

The effect of cerebellectomy on the tonic labyrinth and neck reflexes in the decerebrate cat.

1. Tonic labyrinth and neck reflexes were studied separately and in combination in the decerebrate cat before and after acute cerebellectomy. Reflex effects were observed as changes in the length of the medial head of triceps in both forelimbs. 2. In the decerebrate cat with an intact cerebellum the tonic labyrinth reflexes, elicited by side-up and side-down head rotations, produced asymmetric length changes in the medial head of triceps in both forelimbs, as described by Lindsay, Roberts & Rosenberg (1976). After cerebellectomy, head movements in either direction produced reflex shortenings in the medial triceps in both forelimbs, in contrast to the normal reciprocal reflex length changes. The presence of the cerebellum is thus required for the occurrence of the normal asymmetric labyrinth reflexes. 3. The direction of the neck reflexes remained unchanged by cerebellectomy. 4. It is suggested that the postural disturbances following cerebellar damage may partly result from the change in the form of the tonic labyrinth reflexes, which in combination with the tonic neck reflexes would no longer act in a stabilizing manner on the trunk.

An experimental study of end effects in the extensional deformation of polymers

The sample geometry and the constraints of grips used in measurements of mechanical properties affect the values of the elastic constants obtained. This is known to be more serious for anisotropic materials. Three polymers have been examined in extension: oriented polyethylene terephthalate, isotropic low-density polyethylene, and oriented nylon 6. The results all show a linear increase of compliance with increasing reciprocal sample length. Subtracting the displacements of samples so short that the grips were initially in contact (“zero-length” samples) indicates that the effect is caused mainly by shear in the grips. Extrapolation of the results to zero reciprocal length gives the true sample compliance, and this removes discrepancies in comparing shear compliances derived from extensional and torsional methods.

A simple way of understanding some exact results on critical phenomena in non-homogeneous and finite Ising systems II: Cooperativity in periodically layered systems

By means of a continuum approximation introduced in a preceding paper, we present a more transparent treatment of the critical phenomena in the specific heat of a two-dimensional Ising model with a “layer” type spatial inhomogeneity of its exchange coupling. This calculational procedure (which will be tested by comparison with an earlier exact treatment by Becker and Hahn) enables us to develop a physical picture of cooperative effects in real (spatially non-homogeneous) systems with quenched impurities. To be more concrete: By examination of the one-particle eigenvalues and eigenfunctions of the transfer matrix, it becomes apparent how the critical behaviour is dominated by i) “local” correlation lengths, describing the spatial decay of correlations within homogeneous subdomains of the system and becoming large near certain temperatures which indicate transitions from disorder to spin alignment within each of these domains, and ii), by the spatial average of the inverse of these local decay lengths, termed “global” reciprocal correlation length, which describes correlations between different subdomains on a scale on which these domains look like fluctuating “block spins”. This “global” decay length diverges at a true phase transition of the system as a whole. Correspondingly, for each sort of subdomain a sharp, but finite logarithmic peak in the specific heat and, as long as the layering of the inhomogeneous Ising system is periodic, a true logarithmic singularity are found.