Transport in two dimensions. I the self-diffusion coefficient

The Mori formalism is used to study generalized transport coefficients in two dimensions. All finite multilinear products of the single particle density and momentum density comprise the set of the variables in the calculation of the self-diffusion coefficient. A self-consistent equation, which is a non-linear integral equation, is obtained for the leading asymptotic behavior of the generalized self-diffusion coefficient. An asymptotic solution is presented which for small wavevector (k) and frequency (s) behaves like $\text{In}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(s + k}{}^2\text{D)}{}^{\mathrm{?1}}$, where D has the dimensions of a diffusion coefficient. The mean square and mean fourth displacements of a tagged particle are also calculated. The long time behavior of the momentum correlation function exhibits a tail of the form $\text{[t}\text{In}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(t)]}{}^{\mathrm{?1}}$ whose coefficient is dependent of the intermolecular potential.     

Method for studying metal ion self-diffusion in some inorganic compounds

A method for studying metal ion self-diffusion in oxides (or other inorganic compounds) is described. The method involves oxidation of an appropriate metal to form a dense, single-layered scale of the lowest valent oxide (e.g. MnO on Mn). The specimen is then treated in high vacuum, and the evaporation of metal diffusing through the scale is measured. From the rates of metal diffusion/evaporation as a function of scale thickness information about the defect structure is obtained. The metal ion self-diffusion coefficient is determined from the rate of metal transport (evaporation) through a scale with known thickness. The requirements and limitations of the method are discussed. The use of the method is illustrated for Mn self-diffusion in MnO at 1100°C. The self-diffusion coefficient of Mn in MnO is proportional to the square root of the oxygen pressure, , in t MnO phase field near the MnO/Mn₃O₄ phase boundary. It is also tentatively concluded that the predominating defects near the Mn/MnO phase boundary are manganese interstitials.     

Etude de l'autodiffusion sous pression hydrostatique dans la phase cubique a faces centrees du cerium

The self-diffusion coefficient increases under hydrostatic pressure in γ f.c.c. cerium as in δ b.c.c. cerium. The volume of activation for self-diffusion is negative: $\text{ΔV = ?3}\text{cm}{}^3\text{mol}$ or ?15% mol. vol.These results are in good agreement with Nachtrieb's empirical correlation between the volume of activation and melting point.     

Exact quantum Sine-Gordon soliton form factors

Soliton form factors are constructed using Zamolodchikov's proposed exact massive Thirring model S-matrix. The asymptotic behaviour of the electromagnetic form factor is $\text{～(?t)}{}^{\mathrm{<mtext>?</mtext><mtext>g</mtext><mtext>2</mtext>}}$. The quasiclassical limit is not the previously accepted result.     

Interacting partons

By analysing the asymptotic form of Feynman diagrams in quantum field theory a modified parton model is proposed, which takes into account the parton interaction leading to the breaking of the Bjorken scaling for $\text{Q}{}^2\text{? 20?30 (}\text{GeV}\text{/c)}{}^2$.     

Etude de l'autodiffusion dans la phase cubique centree de l'europium et du gadolinium

The self-diffusion coefficient follows a relation of the form : D = (1,0_?0.4^+0.7)exp (?shape=case>34400RT±700)

$\text{cm}{}^2\text{sec}$

for b.c.c. europium D = (1,0_?0.3^+0.5) × 10^?2 exp(?

$\text{32700 ±4000}\text{RT}\text{)}$

$\text{cm}{}^2\text{sec}$

for β-b.c.c. gadolinium.Whereas europium has normal self-diffusion parameters, β-b.c.c. gadolinium must be set in the class of the anomalous b.c.c. rare-earth metals.From these results we conclude that there exists no evident connexion between the instability of the 4f shell and the activation energies anomalously low in the b.c.c. phases of the rare-earth metals.     

Statistical flow-oscillator modeling of vortex-shedding

The very important engineering problem of modeling the fluid-structure interaction occurring during the shedding of vortices has defied, and will probably continue to defy, a closed form exact solution for the foreseeable future. Therefore, an attempt must be made to extract relevant information about the process in order to be able to have a basic understanding of it for the purpose of analysis. A useful method involves the flow-oscillator concepts of Hartlen and Currie [1] redefined here for stochastic processes. The fluid-structure system is assumed to be governed by the cross-coupled equations

$\text{x}\text{?}\text{(t)+2ξω}{}_{\mathrm{n}}\text{x}\text{?}\text{(t)+ω}{}^2{}_{\mathrm{n}}\text{=C}{}_{\mathrm{e}}\text{(t)pV}{}^2{}_0\text{(t)DL/2m (i)}$

$\text{C}\text{?}{}_{\mathrm{e}}\text{(t)+{α ? βC}{}^2{}_{\mathrm{e}}\text{(t)+γC}{}^4{}_{\mathrm{e}}\text{(t)}}\text{C}\text{?}{}_{\mathrm{e}}\text{(t)+ω}{}^2{}_0\text{C}{}_{\mathrm{e}}\text{(t)=b}\text{x}\text{?}\text{(t), (ii)}$

where these equations govern the structure and fluid oscillators, respectively. The fluid damping is non-linear. These equations are taken as stochastic differential equations because of the many unpredictable, random effects that determine the loading and response. The lift coefficient C_$\text{l}$(t) is assumed to be a zero mean, narrow band process and the velocity V₀, composed of a uniform, constant velocity current plus oscillating wave, a broad band process. The analysis is based on solving equation (i) for x(t) by using Duhamel's integral and substituting its derivative $\text{x}\text{?}\text{(t)}$ into equation (ii). This equation is then used to derive the Fokker-Planck equation for the process C_$\text{l}$(t). To obtain the Fokker-Planck equation, slowly varying variables are replaced by their long-time averages [2] and then the method of stochastic averaging is employed [3, 4]. The moment equation for the lift-oscillator process is derived from the Fokker-Planck equation and, as equation (ii) is non-linear, one finds the moment equation to be in terms of higher order moments. A truncation scheme [5] is used to derive the moment generating function. It is possible then to generate the first and second order statistics of the lift coefficient and the structure response in terms of the empirical parameters of fluid damping. This work was carried out in conjunction with an analysis of ocean wave-current forces with application to offshore fixed structures [6].     

The Dyson instability and asymptotics of the perturbation series in QED

The probability of the ground state decay at e² < 0 is calculated by the steepest descent method in fermionic electrodynamics. The saddle points are the solution of some equations which have been obtained by calculating the asymptotic of the Dirac operator determinant in a very strong external field. The SO(3) × O(2) solutions are found explicitly. The main contribution from them to the Nth coefficient of perturbation theory is proportional to $\text{(?α)}{}^{\mathrm{<mtext>N</mtext><mtext>2</mtext>}}\text{Γ(}\text{N}\text{2}\text{)}$, where $\text{S = 2}{}^2\text{3}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}\text{π}{}^3$.     

Memory effects in the diffusion of an interacting polydisperse suspension: II. Hard spheres at low density

We consider the diffusion of a low density suspension of polydisperse hard spheres. In a previous article (I) we have obtained at long wavelength a general expression to first order in density for the memory matrix appropriate to such a system. In the present article we evaluate the memory matrix in closed form using a certain approximations for the hydrodynamic interaction and for the 2-body propagator. We find that memory effects convert the exponential decay of density correlation functions to long time power law decay of the form $\text{t}{}^{\mathrm{<mtext>-</mtext><mtext>5</mtext><mtext>2</mtext>}}$. We consider the so-called long time self-diffusion constant and show that memory effects to first order in density are negligible compared to the first cumulant contribution. Our treatment shows that for hard sphere systems it is essential to treat the short distance hydrodynamic forces accurately.     

Theory of the self-diffusion (spin diffusion) coefficient of an electron gas

The self-diffusion (spin-diffusion) coefficient D of an electron gas is investigated by means of a proper connected diagram expansion which treats a collision in a medium (rather than in vacuum) in a natural manner. By calculating real and virtual interaction processes to the order e⁴ (second order in interaction strength), we obtain

for a non-degenerate gas where n₀ represents the density. The square-root density (n₀) dependence is noteworthy. In the calculation, no cut-off limiting low momentum transfer in collision is introduced; the screening effect which is important at higher densities, is neglected.     

Chemical diffusion in nonstoichiometric chromium sulfide,Cr2+yS3

Using the re-equilibration kinetic method the chemical diffusion coefficient in nonstoichiometric chromium sesquisulfide, Cr_2+yS₃, has been determined as a function of temperature (1073–1373 K) and sulphur vapour pressure (10?10⁴ Pa). It has been found that this coefficient is independent of sulphur pressure and can be described by the following empirical equation: $\text{D}\text{?}\text{Cr}{}_{\mathrm{2+y}}\text{S}{}_3\text{=50.86}\text{exp(-39070 cal/mole/}\text{RT)}\text{(cm}{}^2\text{s}{}^{\mathrm{?1)}}$. It has been shown that the mobility of the point defects inCr_2+yS₃ is independent of their concentration and that the self-diffusion coefficient of chromium in this sulfide has the following function of temperature and sulphur pressure: . (cm²s^?1).     

A note on the momentum distribution function for an electron gas

The one-electron momentum distribution function $\text{〈a}{}_{\mathrm{k\sigma }}{}^2\text{a}{}_{\mathrm{k\sigma }}\text{〉}$ for an electron gas is investigated by a diagrammatic analysis of perturbation theory. It is shown that $\text{〈a}{}_{\mathrm{k\sigma }}{}^2\text{a}{}_{\mathrm{k\sigma }}\text{〉}$ has the following exact asymptotic form for large k (k ? p_F; p_F, the Fermi momentum): $\text{〈a}{}_{\mathrm{k\sigma }}{}^2\text{a}{}_{\mathrm{k\sigma }}\text{〉 =}\text{4}\text{9}\text{(}\text{αr}{}_{\mathrm{s}}\text{π}\text{)}{}^2\text{×(}\text{p}{}_{\mathrm{<mtext>F</mtext>}}{}^8\text{k}{}^8\text{) g?(0) + ?}$, where g?(0) is the zero-distance value of the spin-up-spin-down pair correlation function. The physical implications of the above asymptotic form are discussed.     

Shell model calculations of inelastic electron scattering from 51V and effective charges

Form factors for inelastic electron scattering for the excitation of the $\text{3}\text{2}{}^{\mathrm{?}}$, $\text{5}\text{2}{}^{\mathrm{?}}$, $\text{9}\text{2}{}^{\mathrm{?}}$, $\text{11}\text{2}{}^{\mathrm{?}}$ and $\text{15}\text{2}{}^{\mathrm{?}}$ members of the ($\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}\text{)}{}^3$ configuration in ⁵¹V are studied by considering the effect of configuration mixing in the shell model. The inclusion of the highly excited configurations which does not contribute to the γ-transitions is shown to give rise to significant modifications of the form factors in large momentum transfer regions. Under the assumption of the static central potential as the residual interactions between protons, good agreement with the experimental data on form factors is obtained by the ordinary force as well as smaller effective charges than the bare charge for higher multipole transitions.     

On a least upper bound Tc of superconductors with paramagnetic impurities

We have obtained a least upper bound, $\text{k}{}_{\mathrm{B}}\text{T}{}_{\mathrm{c}}\text{? c(μ}{}^1\text{, t)A}$, on the critical temperature T_c of an isotropic superconductor with paramagnetic impurities described by the scattering matrix t for fixed values of μ¹. We have also obtained the corresponding optimal spectrum $\text{α}{}^2\text{F(m) = Aδ[ω?d(μ}{}^1\text{, A]}$. The numerical results for the functions $\text{c(μ}{}^1\text{, t)}\text{and}\text{d(μ}{}^1\text{, t)}$ are presented for $\text{α}{}^1\text{= 0.1}$ and 0.16 in the form of universal curves representing $\text{c(μ}{}^1\text{, t)}\text{and}\text{d(μ}{}^1\text{, t)}$ as functions of the reduced impurity concentration $\text{t}\text{= t/A}$. We have also established an upper limit to the reduced critical concentration $\text{t}{}_{\mathrm{crit}}$ for an arbitrary shape of α²F(ω)1.     

Capillary waves on free liquid surfaces: A study of the role of the intermolecular potential in the dispersion

A classical hydrodynamic study of the dispersion of ripplons at the free surface of a liquid has been made. It is shown that the equations may be written so as to directly reveal the role of the intermolecular potential in determining the form of the dispersion relation.For an incompressible fluid, the coefficients in the dispersion relation are just the moments of an averaged intermolecular potential. Further general analytic results were also derived showing how surface phonon (ω ∝ κ) and ripplon ($\text{ω ∝ κ}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$) type relations arise and why the ripplon dispersion relation necessarily describes a transverse excitation.     

Light-cone expansions,the parton model and virtual γγ scattering

It is shown that it is sufficient to use the light-cone algebra of currents and the algebra of bilocal operators to find the asymptotic behaviour of the γγ scattering amplitude when one (or two) of the photon masses q_1,2² is large, and for an arbitrary value of the energy squared s = (q₁+q₂)². A general form of this asymptotic behaviour is obtained. The box-diagram is dominant over the wide region in $\text{s(μ}{}^2\text{« s « q}{}_1{}^2\text{q}{}_2{}^2\text{/μ}{}^2\text{,μ ～ 1}\text{GeV}\text{)}$ and so the asymptotic amplitude is known completely. It is shown that the parton model of the type of ref.[8] gives the same predictions for the asymptotic behaviour of the γγ amplitude.     

Ellipsometric investigation of the skeletonization process of langmuir-blodgett films

When Langmuir-Blodgett films, consisting of a mixture of a long chain carboxylic acid and its salt are immersed in benzene, the acid dissolves, while the salt remains as a skeleton. The density of the film is lowered in this way and so is the refractive index. The process can thus be followed by performing ellipsometric measurements as a function of time of skeletonization. It is established that skeletonization is, in first instance, a diffusion-controlled process with a diffusion coefficient of about $\text{5 × 10}{}^{\mathrm{?14}}\text{cm}{}^2\text{sec}$ at room temperature. The salt also slowly dissolves at a rate of about $\text{1.6 × 10}{}^{10}\text{molecules}\text{/}\text{sec}\text{cm}{}^2$.     

Role of the effective nucleon mass for pion condensation in nuclear matter

The dependence of the threshold for pion condensation in symmetric nuclear matter on the effective nucleon mass $\text{M}{}^1$ is investigated, using extrapolations of $\text{M}{}^1\text{(?)}$ to high densities ?, based on boson exchange mechanisms. It is found that if $\text{M}{}^1\text{(?)}$ decreases below the standard value $\text{M}{}^1\text{=0.8M}$ as the density increases beyond the nuclear matter density ?₀, the critical density is raised considerably beyond ?₀ once short-range repulsive correlations are included.     

The equivalent potential for relativistic confined systems

In previous papers a method of obtaining bound states and wavefunctions for confined relativistic systems was presented. The input is the asymptotic expansion of the two-point functions. Confinement is imposed by systematic removal of the two-particle cut. We extend this method by developing an equivalent (angular momentum dependent) potential, which gives the correct wavefunction to a given order of R, the infrared scale parameter. We prove the uniqueness of the wavefunction by requiring that there are no CDD poles and by the connection of our moment conditions to the requirement that the residues of the bound-state poles must be positive. Finally we test the bound-state approximation for a system defined by an equivalent potential $\text{V(r) = λ}{}^2\text{tanh}{}^2\text{(}\text{g}{}^2\text{r}\text{λ}\text{)}$. Although in this case there is a threshold we still find excellent results when $\text{λ}{}^2\text{g}{}^2$ is large, i.c., when there are many bound states.