Breakdown of normal threshold dispersion relation for the vacuum polarization

Stating a concrete example, it is shown that the vacuum polarization develops anomalous singularities. The analytic structure of the vacuum polarization in these circumstances is discussed in some detail. It appears that its large q² behaviour is not linked anymore to the asymptotic form of the e⁺e^? total cross section. Since it is the large q² behaviour of the vacuum polarization which solely can be calculated in asymptotically free gauge theories (giving in leading order the naive quark model result), this is a warning to using the naive quark model result for the e⁺e^? total cross section as well in fitting the data.     

Die Nusseltsche Zahl für laminare Strömung im zylindrischen Rohr mit konstanter Wandtemperatur

Summary The Nusselt modulus for laminar flow through a pipe with constant temperature of the walls has been determined as a function of the Peclet modulus. This function is represented by a diagram.     

Interaction of vortices in type II superconductors and the behavior nearH c1 at arbitrary temperature

The interaction between well-separated Abrikosov vortices is calculated using Eilenberger's version of the Gorkov theory. Let λ and ξ be the decay lengths determining the asymptotic behavior of the magnetic field and the order parameter for an isolated vortex. Then the onset of attraction in the limit of large vortex separation is either given by λ=ξ or by λ becoming complex, whichever occurs first. It is shown that in general a first order transition atH _c1 takes place already in the region of asymptotic repulsion because of the appearance of a minimum in the interaction energy at finite vortex separation.     

Growth kinetics of solvent crazes in glassy polymers

The kinetics of craze growth from sharp cracks in polystyrene (PS) and poly(methyl methacrylate) (PMMA) in contact with liquid methanol were measured with time-lapse photography as a function of the stress intensity factor K_I. At high K_I the craze length in both systems increases as √t if the sides of the craze are protected from methanol and as t if they are not, where t is the elapsed time after loading. If such a side-protected craze is dried under load and then methanol is reintroduced to the crack tip, the methanol front advances with the same kinetics as the original craze growth. This experiment Proves that solvent crazing velocities are limited by the hydrodynamic transport of solvent through the porous craze structure under a capillary pressure driving force (which can be as high as 100 atm). An improved model of fluid flow through the craze is developed and shown to predict craze growth kinetics in good agreement with those observed. The hydraulic permeability of methanol crazes in PS was found to be independent of craze length at small craze length and to be independent of K_I except at very low K_I. Although increases in molecular weight in the range M_w = 200,000 to M_w = 670,000 do not markedly affect the crazing kinetics, they greatly increase the time to fracture of the craze.     

Effect of molecular entanglements on craze microstructure in glassy polymers

Thin films of ten glassy polymers are bonded to copper grids and strained in tension to produce crazes, which are then examined in the transmission electron microscope. The average craze fibril extension ratio λ for each polymer is determined from microdensitometer measurements of the mass thickness contrast of the crazes. The extension ratio λ is found to increase approximately linearly with the chain contour length l_e between entanglements, as determined from melt elasticity measurements of the entanglement molecular weight of these polymers. These results are analyzed by comparing them with λ_max, the maximum extension ratio of an entanglement network in which polymer chains neither break nor reptate (i.e., permanent entanglement crosslinks are assumed). The values of λ_max are given by l_e/d where d, the entanglement mesh spacing in the unoriented glass, is computed from d = k(M_e)^1/2 with k determined either from small-angle neutron scattering results on isolated chains in the glass or from coil size measurements in dilute solutions of a θ solvent. The craze extension ratios fall somewhat below λ_max at low λ but increase to well above λ_max for polymers with high l_e. This comparison suggests a significant contribution due to chain breakage (or reptation) in the higher-λ crazes of large-l_e polymers, which may arise from the higher true stresses in the craze fibrils (which for a given applied stress increase proportionally to λ). The results also imply that a useful way to increase the “brittle” fracture stress and decrease the ductile-to-brittle transition temperature of a glassy polymer is to decrease its entanglement contour length l_e.     

Recoupling coefficients of the symmetric group for shell and cluster model configurations

Recoupling coefficients of the symmetric group are defined from the reduction of irreducible representations of this group to direct product representations of particular subgroups. These recoupling coefficients are analysed and shown to be identical to the invariants of unitary groups arising from the reduction of Kronecker product representations. The recoupling coefficients have applications in configurations of several shells or clusters.     

X-ray mapping of freeze-dried cryosections from biological cells

Elemental mapping of freeze-dried cryosections from biological cells was done using the energy window mode and the quantitative mode. Quantitative mapping requires long measuring times and high electron exposure resulting in inhomogeneous shrinkage and distortion of the cryosection. Collection of X-ray spectra for quantitative analysis of intracellular structures and compartments combined with energy window mapping was found to be sufficiently fast and reliable to reveal inhomogeneities of the element distribution in cells. However, this method is suitable only for mapping elements without overlapping peaks in the X-ray spectrum.     

Craze fibril structure and coalescence by low-angle electron diffraction

Brown has shown that low-angle electron diffraction (LAED) may be used to determine fibril diameters D and spacings D₀ of crazes in thin polymer films. He found, however, that the D and D₀ determined for air crazes in polystyrene (PS) thin films were larger by about a factor of 3 than those in PS bulk crazes determined by using small-angle x-ray scattering (SAXS). We have repeated Brown's LAED experiments and find that the discrepancy may be caused by an aging effect. Our fresh crazes have D and D₀ values from LAED that are comparable to those of bulk PS crazes determined by SAXS. As the craze ages, however, fibrils retract and coalesce in wide regions of the craze, leading eventually to an observable “skin.” Aged crazes thus have much larger D and D₀ values than do fresh crazes. The large molecular mobility of the PS molecules in the fibrils necessary for this aging to occur at room temperature has important implications for fibril failure.     

Pattern propagation in nonlinear dissipative systems

We discuss the problem of pattern selection in situations where a stable, nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous region. Our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known; and we point out that these known properties are consistent with a marginal-stability hypothesis that has been suggested in the theory of dendritic crystal growth. We then describe a more general interpretation of the marginal-stability hypothesis and, finally, present numerical evidence for its validity from three different pattern-forming models.     

On the theory of collective motion in nuclei. III. From Hamiltonian dynamics to vortex-free fluidity

It is assumed that the Hamiltonian for collective motion in nuclei is invariant under the orthogonal group O(n, $\text{R}$). For degenerate orbits in phase space it is shown that the classical Hamiltonian equations reduce to the equations of a vortex-free fluid with a velocity field determined by independent equations of motion.