Exponential Regime Research Articles

Variational solution of the wave equation for a high gain FEL and a finite wiggling radius

The driving current for an FEL amplifier operating in the exponential growth regime is derived for a frequency close to the reasonance one and for a finite value of the wiggling radius of the electrons. The wave equation is solved by applying the generalized Rayleigh-Ritz variational principle. In particular, it is shown that the dispersion equation for the propagation constant can be put into the same form as that derived in absence of wiggling.

Transport of Indicator Clasts by Ice Sheets and the Transport Half-Distance: A Contribution to Prospecting for ORE Deposits

The abundance (A) of granitic and metamorphic clasts in Wisconsinan till in the Dummer Moraine at the edge of the Precambrian Shield of Ontario decreases exponentially but then remains nearly constant at about 4% for more than 800 km down-ice in southwestern Ontario, Ohio, and Indiana. The exponential decrease is attributed to the transport and deposition in the basal ice of the Laurentide ice sheet after it crossed the contact between the granitic rocks of the Precambrian Shield and the overlying Paleozoic carbonate rocks. The transition from exponential to linear distribution was caused by a change in transport mechanism from basal to englacial transport of clasts that were moved above the base of the ice sheet and thus escaped attrition and deposition in basal till. The transport half-distance of a boulder train can be estimated from a plot of In A vs. transport distance within the exponential transport regime. The half-distance can then be used to determine the transition distance, which indicates how...

Theoretical study of FEL active guiding in the small signal regime

We use a self-consistent small signal FEL field equation to study active guiding. We show that self-similar mode propagation is possible only in the exponential regime. These modes must be either growing or decaying; the purely oscillatory self-similar mode is not confined. By a Gaussian approximation, we investigate the eigenvalue structure of the fundamental mode analytically, and derive a cubic equation for the complex growth rate, the laser mode size, and the radius of the phase front curvature.

Generalized gain-spread relation and optical guiding effect in high-gain free electron laser amplifiers

We derive a generalized gain-spread relation for constant-wiggler free electron laser amplifiers which is valid for both low-gain and high-gain devices. It connects the energy spread of the electrons with the intensity and with the phase variation of the radiation field. This relation is discussed in the small-signal and in the exponential gain regimes. In the small-signal regime it reduces to Madey's theorem. In the exponential gain regime it is related to the effect of optical guiding of radiation in the electron beam. We demonstrate analytically and numerically that refractive guiding can take place near the wiggler entrance before gain guiding becomes appreciable.

Nonlinear corrections in a high-gain free-electron laser

A perturbation series in powers of the radiation field strength is developed in order to calculate nonlinear corrections to the high-gain exponential amplification regime of a single-mode free-electron laser (FEL). In such a series, contributions to the optical power gain come only from the odd order terms. We give an estimate of the radiative power threshold at which the lowest-order nonlinear (3rd-order) contribution becomes comparable to the linear-order result.

Nonlinear-response theory of Ising-like systems

The nonequilibrium response of a two-dimensional, paramagnetic system to a rapid and arbitrarily large change (quench) in a symmetry-breaking field is investigated. The complete time dependence of the conjugate order parameter is calculated with a real-space renormalization-group (RG) method. The nonlinear relaxation "rate" is found to be time dependent. Its approach to the corresponding prediction of linear-response theory is characterized. The nonlinear response is found to go over to linear response via an early, exponential regime followed by a crossover to an asymptotic, algebraic approach to linear response at long times. The initial transient decay is quench-strength dependent in contrast with the final behavior, which is largely "universal" and quench insensitive. We study a nearest-neighbor Ising model in a magnetic field in the paramagnetic, "disordered" phase. The dynamics are controlled by a kinetic Ising model with simultaneous spin-flip (adsorption-desorption) and spin-exchange (diffusion) kinetics. The model is applicable to non-equilibrium "dosing" experiments on adsorbed systems. Diffusion is shown not to affect the growth rate of the total order parameter (magnetization or coverage). General recursion relations for the temperature and magnetic field are derived for the entire thermodynamic plane. The RG analysis reproduces linear-response theory for arbitrary magnetic field and is consistent with known scaling laws concerning nonlinear relaxation. The time-dependent magnetization in the critical region has a power-law regime before crossing over into its final exponential decay.

Absence of statistical fluctuations in quasi-one-dimensional disordered systems

The fact that the inverse localization length is normally distributed in disordered onedimensional systems, combined with the delocalizing effects of inelastic scattering, imply that the resistance of thin metallic wires will be a well-behaved random variable in any temperature regime which appears to be accessible by present experimental techniques. In particular, it is shown that the exponentially large fluctuations in the zero-temperature dc resistance found in studies of the Landauer formula can only be of importance in the exponential hopping regime. A condition for these fluctuations to be observed is derived.

Anderson Localization in Two Dimensions

The conductance for a two-dimensional tight-binding model with on-site disorder is calculated numerically with use of the Kubo formula. For weak disorder logarithmic localization is observed, in agreement with the scaling theory. The magnetoresistance is found to be negative in both the logarithmic and exponential localization regimes. Results for a model with random complex hopping matrix elements are also presented.

Chain-branching and initiation rates measured by spatially integrated light emission during reflected shock-wave ignition

Experimental study of ignition kinetics in H2−O2−CO−Ar mixtures by time-resolved photoelectric measurement of the growth of spatially integrated CO−O recombination radiation in reflected shock waves has been extended to cover the ranges 0.1≤H2:O2≤10.0, 1000°≤T≤2500°K, by confining the reactive gas to the downstream end of the shock tube and increasing sensitivity through a large solid angle optical view. The induction period regime of exponential growth of emission intensity, I(t)≈I0*exp(α*t), yields precise values of α* whose dependence upon [H2], [O2], and T determines, for the reactions H + O 2 → k 1 O H + O O + H 2 → k 2 O H + H O H + H 2 → k 3 H 2 O + H the functions: k1=8.6×108 exp {−(12.3±1) kcal/mole (1/T−1/1600)/R} liter/mole/sec and k2k3=1.5×1019 exp {−(20±4) kcal/mole (1/T−1/1600)/R} liter2/mole2/sec2, and indicates that k2 and k3 are, at most, about a factor of 3 from each other, in the present temperature range. The early growth of I(t) prior to the exponential regime, now observed directly, and the parameter I0*, now fairly precisely measured, bear upon the kinetics of chain initiation but require further investigation.

Theory of Unstable Particles

A new method in the theory of unstable particles is introduced. It is applied in this paper to a simple model to show how the exponential regime of the decay can be isolated systematically. It is further shown that our method prescribes the precise conditions to which the initial wave function must be submitted for this exponential decay to ensue. The prescription of these conditions constitutes in fact a definition of an "unstable particle" in quantum theory.