Thomas Erber

Equilibrium configurations of N equal charges on a sphere

The emergence of new levels of complexity that often accompanies the transition from few- to many-body systems is clearly illustrated by the progression of equilibrium states of N charges on the surface of a sphere as N increases. The characteristics of these electrostatic equilibrium states in the range 2<or=N<or=65 can be examined in detail by equilibrating 1000 randomized initial configurations for every value of N. As N increases, the equilibrium states undergo a succession of structural changes. For instance, a state with non-zero dipole moment appears at N=11, an enantiomeric or mirror image state appears at N=15, and a robust metastable state appears at N=16. For values of N exceeding 50, clusters of four or more metastable states with energies within 0.01% of each other are the dominant pattern. In analogy with some other complex systems, these energetically similar states have strikingly different asymmetric configurations.

Classical and quantum theory of synergic synchrotron-Čerenkov radiation

Abstract The power spectra of Cerenkov, synchrotron, and synchrotron-Cerenkov radiation are calculated both classically (by source methods) and quantum mechanically (by mass operator methods). The synergic features of the synchrotron-Cerenkov radiation are pointed out. The existence of a transition region near nβ = 1 [n(ω, H) = index of refraction of the medium; β = v c , v - velocity of the charged particle] coupled with the intrinsic dispersion of the medium is then applied to the discussion of the suppression of X-ray radiation, the construction of X-ray counters, the detection of the quantum corrections, and the modification of the synchrotron radiation from pulsars.

A general phenomenology of hysteresis

Abstract Studies of magnetic cooperative systems have shown that some forms of hysteresis are due to the existence of points of instability, i.e., Hessian singularities, in phase space. For extremely complex systems—in particular, macroscopic systems—these singularities become very numerous and it is possible to derive quantitative conclusions regarding the hysteresis which they generate by purely statistical means. Specifically, we show that both the Rayleigh cube law of energy dissipation in ferromagnetic hysteresis and Doreys damping rule for stress-strain hysteresis are simple consequences of the statistical properties of these phase-space singularities. Since the detailed nature of the interactions which give rise to these singularities is irrelevant in statistical arguments, the application of the cube law can be extended to many other systems which exhibit an asymptotic regime of hysteresis. On a parallel level of generality we show that for virgin hysteresis the cubic law of energy dissipation goes over into a simple linear relation. These features are illustrated in detail with an analysis of structural shakedown. Further properties of hysteresis can be deduced from the shape of the phase-space hypervolumes which enclose the singularities, and from the topology of the dynamical trajectories by which they are linked. In particular, the onset of hysteresis and the evolution of virgin hysteresis into asymptotic hysteresis can be interpreted simply and quantitatively in terms of the crossing of various “band-edges” (≡ boundaries of hypervolumes) in phase space. The well-known antinomy that iterated hysteresis sometimes tends to stabilize (shakedown of structures, training of superconducting magnets), and sometimes tends to destroy (structural failure due to incremental collapse, fatigue) can formally be resolved in terms of graph-theoretical properties of the network of hysteresis trajectories.

The gamma function inequalities of Gurland and Gautschi

Abstract The two majorants in question are Gurlands Inequality [1] and Gautschis Inequality [2] By combining these inequalities it is possible to improve iteratively upon Gautschis result (2) and also to develop upper bounds for Γ(n+²/Γ(n+2²), n = 1,2, … , 0 < ² < l.

Experimental aspects of synchrotron-Čerenkov radiation☆

Various features of synchrotron-Cerenkov radiation are illustrated in the context of the following situations: (1) The passage of high energy electrons through gases, liquids, solids, and plasmas in the presence of magnetic fields. (2) The suppression of synchrotron X-rays from high energy electrons in the earths atmosphere and extraterrestrial environs. (3) The detection of the real part of the Delbruck scattering amplitude. (4) Vacuum polarization effects on radiation by ultra-high energy electrons in intense magnetic fields. (5) Synchrotron-Cerenkov radiation by charged particles heavier than electrons.

Piezomagnetism and fatigue: II

The cumulation of damage in test specimens of AISI 1018 steel, subjected to repeated cycles of tension and compression leading to fatigue failure at Nf cycles, is correlated with the evolution of stress–strain (σ − e) hysteresis and piezomagnetic (B–e) hysteresis. Specifically, the σ − e hysteresis loop areas, when plotted as a function of the number of loading cycles N, show systematic variations that can be identified with the three principal stages of fatigue: initial accommodation (i.e. strain softening or hardening), N < N2; accretion of damage, N2 < N < N3, and terminal failure (crack coalescence and growth); N3 < N < Nf. Data from 49 fatigue trials, spanning the range 1219 ≤ Nf ≤ 250 200, show that the transitional cycles N2 and N3 have an approximately invariant relation to final fatigue failure at Nf: i.e. N2/Nf ≈ 12% and N3/Nf ≈ 90%. Piezomagnetic hysteresis develops in parallel with stress–strain hysteresis and also exhibits transitions at N2PM and N3PM corresponding to N2 and N3. Detailed analyses of eight fatigue trials yield the approximately invariant ratios N2PM/Nf ≈ 12% and N3PM/Nf ≈ 93% where 3561 ≤ Nf ≤ 189 629.

Quantum modifications in magnetic bremsstrahlung

Abstract The relativistic classical expression for the magnetic bremsstrahlung (synchrotron) spectrum of electrons can be generalized to include first-order quantum corrections by the replacement (I(ω) ∼ κ(ω/ω0) → κ((ω/ω0)[1 + (ħω/E)]) where κ ( z ) = z ∫ s ∞ d x K 5 / 3 ( x ) , and ωc is related to the electron energy and magnetic field intensity by ωc(keV) ⋍ 0.06 E2 (GeV) H(kG). These first-order shifts are of negligible significance unless E (GeV) Prompted by the results of a megagauss bremsstrahlung experiment, we have recalculated the synchrotron process including all relevant second-order quantum corrections. In the range the analysis reduces to the simple correspondence I(ω) → κ((ω/ωc)[1 +ħω/E][1 − (ħω/2mc2)2]3/2), which exhibits the surprising feature that the second-order terms can be more significant than the first-order corrections. In fact whenever E2 (GeV) the general results indicate that the spectrum is drastically altered by quantum effects. Since the second-order terms are also linked with an enhancement of the magnetic trident production rate, the matrix elements are evaluated with sufficient generality to allow for inner bremsstrahlung processes.

Hooke's law and fatigue limits in micromechanics

Scanning tunnelling microscopes can be used to detect transitions between reversible and irreversible deformations of materials. Since the occurrence of stress-strain hysteresis is a necessary condition for the generation of material defects, and the cumulation of defects is, in turn, the underlying cause of fatigue failure, the observation of non-hysteretic reversible deformations extending over many atomic lengths implies that mechanical nanoscale devices are potentially capable of having service lives of extremely long duration.

Reversibility, irreversibility: restorability, non-restorability

Simple dynamical systems of point particles are irreversible if their motions cannot be retraced merely by reversing their velocity components. More complicated systems, such as those that exhibit steady hysteresis under the cyclic action of external influences, may be locally reversible, globally irreversible, and yet traverse an ordered set of recurrent states. In still more complex situations incorporating memory dependences, hysteresis effects are generally evolutionary or non-recurrent: nevertheless, in special circumstances, it may be possible to achieve a restoration of some prior states. An example based on the behaviour of an elastic-perfectly plastic torsion spring shows that such restorations require processes that are qualitatively more complicated than those associated with the original evolution of the systems. This asymmetry in the complexity of processes provides another means for assigning a direction to the arrow of time.

Unified radiation formulae for classical and quantum electrodynamics

There are deep conceptual differences between the classical and quantum mechanical treatment of electromagnetic radiation processes. Nevertheless, it is possible to give a formally unified description of the spectral and angular distribution of radiation in both cases in terms of four-dimensional Fourier transforms of currents. We present parallel derivations of the basic radiation formulae utilizing classical electrodynamics as well as spinor quantum electrodynamics. In addition both derivations allow for the presence of a medium with an index of refraction. The practical application of these methods is illustrated by calculations of some specific radiation problems.