Pure Exponentials Research Articles

Compact Modeling of Nonideal Trapping/Detrapping Processes in GaN Power Devices

Compact modeling of charge trapping processes in GaN transistors is of fundamental importance for advanced circuit design. The goal of this article is to propose a methodology for modeling the dynamic characteristics of GaN power HEMTs in the realistic case where trapping/detrapping kinetics are described by stretched exponentials, contrary to ideal pure exponentials, thus significantly improving the state of the art. The analysis is based on: 1) an accurate methodology for describing stretched-exponential transients and extracting the related parameters and 2) a novel compact modeling approach, where the stretched exponential behavior is reproduced via multiple <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</i> networks, whose parameters are specifically tuned based on the results of 1). The developed compact model is then used to simulate the transient performance of the HEMT devices as a function of duty cycle and frequency, thus providing insight on the impact of traps during the realistic switching operation.

The role of stellar radial motions in shaping galaxy surface brightness profiles

Aims: The physics driving features such as breaks observed in galaxy surface brightness (SB) profiles remains contentious. Here, we assess the importance of stellar radial motions in shaping their characteristics. Methods: We use the simulated Milky Way-mass, cosmological discs, from the Ramses Disc Environment Study (RaDES) to characterise the radial redistribution of stars in galaxies displaying type I (pure exponentials), II (downbending), and III (upbending) SB profiles. We compare radial profiles of the mass fractions and the velocity dispersions of different sub-populations of stars according to their birth and current locations. Results: Radial redistribution of stars is important in all galaxies regardless of their light profiles. Type II breaks seem to be a consequence of the combined effects of outward-moving and accreted stars. The former produces shallower inner profiles (lack of stars in the inner disc) and accumulate material around the break radius and beyond, strengthening the break; the latter can weaken or even convert the break into a pure exponential. Further accretion from satellites can concentrate material in the outermost parts, leading to type III breaks that can coexist with type II breaks, but situated further out. Type III galaxies would be the result of an important radial redistribution of material throughout the entire disc, as well as a concentration of accreted material in the outskirts. In addition, type III galaxies display the most efficient radial redistribution and the largest number of accreted stars, followed by type I and II systems, suggesting that type I galaxies may be an intermediate case between types II and III. In general, the velocity dispersion profiles of all galaxies tend to flatten or even ncrease around the locations where the breaks are found. The age and metallicity profiles are also affected, exhibiting...[abridged]

Unitary group adapted state-specific multi-reference coupled cluster theory: Formulation and pilot numerical applications

We present the formulation and the implementation of a spin-free state-specific multi-reference coupled cluster (SSMRCC) theory, realized via the unitary group adapted (UGA) approach, using a multi-exponential type of cluster expansion of the wave-operator Ω. The cluster operators are defined in terms of spin-free unitary generators, and normal ordered exponential parametrization is utilized for cluster expansion instead of pure exponentials. Our Ansatz for Ω is a natural spin-free extension of the spinorbital based Jeziorski-Monkhorst (JM) Ansatz. The normal ordered cluster Ansatz for Ω results in a terminating series of the direct term of the MRCC equations, and it uses ordinary Wick algebra to generate the working equations in a straightforward manner. We call our formulation as UGA-SSMRCC theory. Just as in the case of the spinorbital based SSMRCC theory, there are redundancies in the cluster operators, which are exploited to ensure size-extensivity and avoidance of intruders via suitable sufficiency conditions. Although there already exists in the literature a spin-free JM-like Ansatz, introduced by Datta and Mukherjee, its structure is considerably more complex than ours. The UGA-SSMRCC offers an easier access to spin-free MRCC formulation as compared to the Datta-Mukherjee Ansatz, which at the same time provides with quite accurate description of electron correlation. We will demonstrate the efficacy of the UGA-SSMRCC formulation with a set of numerical results. For non-singlet cases, there is pronounced M(s) dependence of the energy for the spinorbital based SSMRCC results. Although M(s) = 1 results are closer to full configuration interaction (FCI), the extent of spin-contamination is more. In most of the cases, our UGA-SSMRCC results are closer to FCI than the spinorbital M(s) = 0 results.

Luminescence Lifetime Standards for the Nanosecond to Microsecond Range and Oxygen Quenching of Ruthenium(II) Complexes

A rapid and reproducible method for determining the temperature dependence of luminescence lifetimes has been developed. With the use of this method, a set of standards for the excited-state lifetime oxygen quenching of several ruthenium(II) transition metal complexes was established. With the use of three solvents of different viscosities and two metal complexes with widely different lifetimes, an overlapping range of ca. 100 ns to 6 micros was obtained. The decays are pure single exponentials, which means that they can be used reliably with both phase and pulsed lifetime instruments. For a pure single-exponential decay, a properly operating phase shift instrument will give the same lifetime as a time domain instrument. With the use of a thermal deactivation model and a three-parameter temperature-dependent oxygen quenching constant, the lifetime temperature-dependent data was well fit by a simple six-parameter equation that covers the temperature range of 10-50 degrees C and oxygen pressures from 0 to 1 atm of oxygen with excellent precision (ca. <1%). This permits both laboratory and field calibration of instruments.

DoubleSlicing: A non-iterative single profile multi-exponential curve resolution procedure : Application to time-domain NMR transverse relaxation data

A new non-iterative curve resolution technique for resolving single decay profiles is proposed. The new technique, called DoubleSlicing, is based on the Decra (Direct Exponential Curve Resolution Algorithm) principle. While the original Decra was designed to resolve several decay curves simultaneously and thus fitting common pure exponentials, DoubleSlicing can resolve single decay profiles by a simple double data transformation followed by an analytical and unique three-way decomposition. The new approach is successfully demonstrated on experimental NMR CPMG relaxation data, measured on combinations of unmixed paramagnetic CuSO 4 solutions. Decay signals of the water component were acquired following an innovative experimental design that ensured no interaction between the components present in each sample under observation. DoubleSlicing proved to be accurate in estimating relaxation times differing in one order of magnitude (range: 19.6–159.4 ms). Its performance was comparable to discrete exponential fitting with the advantage of being much faster – in terms of computation time, DoubleSlicing outperformed exponential fitting by a factor of four.

Spectral functions, maximum entropy method and unconventional methods in lattice field theory

We present two unconventional methods of extracting information from hadronic 2-point functions produced by Monte Carlo simulations. The first is an extension of earlier work by Leinweber which combines a QCD Sum Rule approach with lattice data. The second uses the Maximum Entropy Method to invert the 2-point data to obtain estimates of the spectral function. The first approach is applied to QCD data, and the second method is applied to the Nambu—Jona-Lasinio model in (2+1)D. Both methods promise to augment the current approach where physical quantities are extracted by fitting to pure exponentials.

Synchronous machine parameter determination using the sudden short-circuit axis currents

Small salient-pole alternators are being embedded in power systems in growing numbers and parameter sets which accurately reflect the transient response of such machines are in demand. High levels of local saturation and rapidly changing flux densities result in transient currents that are far from the pure exponentials of classical theory so that estimation by curve fitting is difficult. A new approach, which encourages designer intervention, is described and this is seen to be reliable and repeatable, with a good quality of fit, even in the sub-transient region.

B and R Surface Photometry of Faint Galaxies in the Area of Three Cosmic Voids

We performed B and R surface photometry of 92 faint mostly late type galaxies, selected towards three nearby voids (Hopp et al., 1995 A&amp;AS 109, 537). CCD frames were taken with the Calar Alto 3.5m telescope and with the MPIA/ESO 2.2m telescope. We calculated the azimuthally averaged equivalent profiles, asymtotic total magnitudes and colours, and performed ellipse-fitting of isophotes. Observed light profiles were fitted with a power law μ = μ0 + 1.086αr1/n. About 40% of profiles are pure exponentials (n = 1); 38% of the profiles reveal a bulge/disk or disk/disk composition; 20% of them show central light depression. We classified the nearby galaxies (z ≤ 0.04) as being related either to clusters (11 galaxies), to sheets (23) or to voids (9). The obtained parameters are compared to those of HSB and of LSB field galaxies.

Approximation by Piecewise Exponentials

A function is called an exponential if it is a linear combination of products of polynomials with pure exponentials. In this paper lower and upper bounds for families of spaces of piecewise exponentials are established. In particular, the exact $L_p $-approximation order $(1 \leq p \leq \infty )$ is found for a family $\{ {S_h } \}_{h > 0} $ of function spaces when each $S_h$ is generated by an exponential box spline and its multi-integer translates.

Lifshitz tails and long-time decay in random systems with arbitrary disorder

In random systems, the density of states of various linear problems, such as phonons, tight-binding electrons, or diffusion in a medium with traps, exhibits an exponentially small Liftshitz tail at band edges. When the distribution of the appropriate random variables (atomic masses, site energies, trap depths) has a delta function at its lower (upper) bound, the Lifshitz singularities are pure exponentials. We study in a quantitative way how these singularities are affected by a universal logarithmic correction for continuous distributions starting with a power law. We derive an asymptotic expansion of the Lifshitz tail to all orders in this logarithmic variable. For distributions starting with an essential singularity, the exponent of the Lifshitz singularity itself is modified. These results are obtained in the example of harmonic chains with random masses. It is argued that analogous results hoid in higher dimensions. Their implications for other models, such as the long-time decay in trapping problems, are also discussed.

Translates of exponential box splines and their related spaces

Exponential box splines ( E B EB -splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space H \mathcal {H} spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space H {\mathbf {H}} of all entire functions spanned by the integer translates of an E B EB -spline. This investigation relies on a detailed description of H \mathcal {H} and its discrete analog S \mathcal {S} . The approach taken here is based on the observation that the structure of H \mathcal {H} is relatively simple when H \mathcal {H} is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex E B EB -splines. The first is when the Fourier transform of the E B EB -spline vanishes at some critical points, the second is when H \mathcal {H} cannot be embedded in S \mathcal {S} and the third is when H {\mathbf {H}} is a proper subspace of H \mathcal {H} . We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.

Small-time behaviour of quantum nondecay probability and Zeno's paradox in quantum mechanics

A detailed investigation of the small-time deviations of the quantum nondecay probability from a pure exponential is made to study their physical consequences. Specific consideration is given to the problem of the dependence of the lifetime on the characteristics of the measuring apparata whose possible occurrence has recently been pointed out. In particular we investigate the problem of the indefinite increase of the lifetime when the frequency of the measurement processes tends to infinity, an effect referred to as Zeno's paradox in quantum mechanics. It is shown that, if the uncertainty relations are properly taken into account, the arguments leading to the paradox are not valid. Moreover, by the same kind of arguments, it is shown that the dependence of the measured lifetime on the frequency of the measurement processes, even though present in principle, is practically not detectable. To verify experimentally that the reduction process is effective one must then resort to the comparison of an experiment with reductions with one in which no interactions with the environment take place.

Direct observation of rotational diffusion by picosecond spectroscopy

Time dependent fluorescence depolarization measurements have for the first time been extended to the picosecond regime by using streak camera optical multichannel analyzer detection. Fluorescein derivative dyes (M.W. 500-1000) rotate in polar solvents as if their volume is at least double that of the free molecule because of solvent attachment, an effect noted by Einstein and elaborated upon by Marinesco and Perrin many years ago. This effect is apparently the major cause of the breakdown of the literal Einstein hydrodynamic model for rotation of these relatively small molecules in solution. The solvent attachment also very likely reduces the effect of molecular shape on rotational depolarization, causing these molecules to behave more like spheres than their molecular structure would imply. Both the fluorescence decay curve and the rotational correlation function derived from the experimental data, within the limits of experimental error, decay as pure exponentials. In addition, unlike most fluorescence probe experiments on nanosecond time scales applied to the study of macromolecular structure, the theoretical value of 0.4 for the polarization anisotropy at zero time is observed. Comparison of these types of results on picosecond time scales could have implications in the study of flexions and rotations of the substructure of large molecules having biological importance.

Influence of cross-relaxation on NMR spin-lattice relaxation times

When intramolecular dipole-dipole interaction is a significant nuclear magnetic resonance spin-lattice relaxation mechanism, the recovery of the longitudinal magnetization after a perturbation no longer follows a simple exponential curve described by a single time constant T 1. This cross-relaxation effect is analyzed for the case of a nonselective 180° pulse applied to a system of two spin 1 2 nuclei (I and S), in terms of two “external” relaxation parameters, T XI and T xs, and the dipolar relaxation time T 1D. If the experimental relaxation data are plotted on a semilogarithmic graph, the deviations from linearity due to cross-relaxation may not be large enough to be detected in the presence of statistical errors on the measurements, but the derived slope can, nevertheless, show a significant error. In general for weakly coupled systems these errors are found to be quite small; they vanish for the I signals of a heteronuclear system where the S nucleus is strongly irradiated or where T XI or T XS is much shorter than T 1D. Experimental results for the heteronuclear 13CH spin system in formic acid and the homonuclear HH spin system in cytidine indicate errors of the order of 10 %. Cross-relaxation effects can be much more serious in strongly-coupled spin systems with appreciable asymmetry in the external relaxation ( T XI ≠ T XS). The lines of an AB quartet follow four separate recovery curves showing marked deviations from pure exponentials, particularly for the weak outer lines. This is illustrated with experimental results for the ring protons of tyrosine.

Fluorescence decays and phonon processes in the excited states of fluorescent ions

Consider a three-level fluoresent system with levels 1, 2 and 3 indicating the ground state and two excited (metastable) states, respectively. Assume that the system can be optically excited by pumping into an absorption band above level 3 and that the excited centers decay from this band to level 3 by fast radiationless processes. When excited by a fast pulse of light levels 2 and 3 will decay with patterns which depend on the populations at the end of the pulse and the various transition probabilities. These probabilities include phonon processes and are dependent on the temperature of the crystal and the coupling between fluorescent system and host lattice. The conditions under which the excited levels may decay with the same pattern or with such patterns as pure exponentials, fluorescence rise followed by decay, double exponential, are examined. The decays of Cr3+ and Sm2+ in several host lattices and in the (4÷700) °K region are presented and the variations of thetype of pattern are related to the responsible phonon processes. The evidence of upward, phonon annihilation processes in the decay patterns of the5D0 and5D1 states of BaClF:Sm2+ is also presented.

The Wave Form of Luminescence Emitted by Noctiluca

The wave shape, intensity, and time course of the flash were examined with the aid of electronic operations in order to characterize the luminescence response and examine the in vivo dynamics of the light reaction. The most prominent single component of the flash shape is its exponential decay, beginning several milliseconds after the intensity maximum, with a mean rate constant at 23 degrees C of -0.088 msec(-1). Earlier components of the flash curve are more complex, exhibiting no pure exponentials with time. As predicted from previous observations, the time course of the flash triggered by a propagated action potential, and therefore influenced by the conduction time of the triggering potential, is measurably slower than that of the synchronously triggered flash. The time course of emission from individual specimens is otherwise quite stable, undergoing only limited slowing with short-interval fatigue or specimen deterioration in spite of marked changes in the amplitude of the wave form. Relative stability of amplitude is obtained when flashes are elicited at regular intervals greater than 10 sec. On the basis of an analogue computer simulation (Appendix) the dynamics of the luminescence wave shape were found to be compatible with a short sequence of first order processes acting on an initial brief transient.

Fluorescence Studies of Energy Transfer between Single and PairCr3+Systems inAl2O3

The fluorescence intensities and lifetimes of various no-phonon lines in ruby samples with 2.1 and 0.94% ${\mathrm{Cr}}^{3+}$ were studied from 13 to 700\ifmmode^\circ\else\textdegree\fi{}K. The temperature dependences of the intensities of numerous lines were used to determine the energy levels of two pair systems. The relative intensities of the single-ion ($R$) lines and the pair ($N$) lines are proportional to Boltzmann factors ${e}^{\frac{\ensuremath{-}\ensuremath{\Delta}E}{\mathrm{kT}}}$, where $\ensuremath{\Delta}E$ is the difference in energy of the metastable states, at high temperatures but not at low temperatures. The deviation from this law is greater and occurs at higher temperatures in the less concentrated sample. The fluorescence decays of the $R$ and $N$ lines are pure exponentials in the temperature region where the $N$ lines follow a Boltzmann law; at low temperatures the $N$ lines have an initial rise in fluorescence and then a decay. These results are interpreted in terms of coupled fluorescence systems consisting of single ions and pairs. A phonon-assisted cross-relaxation coupling process is suggested which allows for thermalization of the systems at high temperatures and direct pumping of the pair systems by the single-ion system at low temperatures. Energy transfer from single-ion to multi-ion systems was also observed.

Superposition of Configurations and Natural Spin Orbitals. Applications to the He Problem

The method of superposition of configurations is examined in its application to the helium atom in two cases: a 21×21 matrix including all configurations up to 〈6s〉2, and a 20×20 matrix with all configurations up to the 4-quantum level, including angular terms. A new radial limit is established at −2.87900±0.00003, and this is used to discuss the convergence of such expansions in Legendie functions. The variation with scale factor is discussed in detail. The wave functions are analyzed in terms of natural spin orbitals (NSO's), which seem to have many advantages. The first NSO bears a striking resemblance to the Hartree-Fock function, and the first two together provide a close approximation to the solution of the extended Hartree-Fock equations with different orbitals for different electrons. An energy of −2.877924 is obtained for the best (u, v) function found. An analysis of the results suggests that inner orbitals may be better represented by pure exponentials than by Hartree-Fock orbitals whenever additional correlational degrees of freedom are permitted. Expressed in approximate NSO form, the wave function is almost invariant to choice of basis set, provided that the latter is reasonably chosen. In particular, the necessity of including continuum terms along with the discrete hydrogen-like set is demonstrated.

Generalized Fourier Integrals

Classical Fourier analysis is based on the decomposition of time functions into pure sine waves or exponentials. In this paper, a more general method of decomposition involving periodic elementary time functions is developed. Explicit inversion formulas are obtained and a general expression for the inverse kernel is derived. In the special case of elementary signals having the form of square waves, it is found that the coefficients entering into the expression for the inverse kernel are given by a function which is closely related to the Mobius function. As an application, the impulsive response of an ideal filter which passes all square waves whose fundamental frequency is less than \omega_0 and stops all those whose fundamental frequency is greater than \omega_0 , is determined.

A One-Term Pure Exponential Formula for the Spectral Distribution of Radiant Energy from a Complete Radiator*

Get PDF Email Share Share with Facebook Tweet This Post on reddit Share with LinkedIn Add to CiteULike Add to Mendeley Add to BibSonomy Get Citation Copy Citation Text Irwin G. Priest, "A One-Term Pure Exponential Formula for the Spectral Distribution of Radiant Energy from a Complete Radiator*," J. Opt. Soc. Am. 2, 18-22 (1919) Export Citation BibTex Endnote (RIS) HTML Plain Text Citation alert Save article