Nonexponential decay law

The decay of a nonstationary state usually starts as a quadratic function of time and ends as an inverse power law (possibly with oscillations). Between these two extremes, the familiar exponential decay law may be approximately valid. The main purpose of this paper is to find the conditions which must be satisfied by the Hamiltonian and by the initial state, for the exponential law to have a significant domain of validity. It is shown that the evolution of a nonstationary state is governed by a nonnegative function W(E), having the dimensions of an energy. Among its properties are: the energy uncertainty is given by $\text{(ΔH)}{}^2\text{= ?W(E)dE}$, and the inverse lifetime by Γ = 2πW(E₀), where E₀ is the expectation value of H. The detailed shape of W(E) defines two characteristic times between which the exponential decay law is a good approximation: roughly speaking, the smoother W(E), the larger the domain of validity of the exponential law. For instance, if W(E) is very smooth ($\text{|}\text{dW}\text{dE}\text{| ? 1}$) except for a sharp threshold at E = E_thr, the transition from quadratic to exponential decay occurs for $\text{t ?}\text{1}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}$, and the transition from exponential to inverse power law when $\text{Γt ?}\text{log}\text{[}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}\text{Γ}\text{]}$.     

Connectivity of growing networks with link constraints

We propose a growing network model with link constraint, in which new nodes are continuously introduced into the system and immediately connected to preexisting nodes, and any arbitrary node cannot receive new links when it reaches a maximum number of links k_m. The connectivity of the network model is then investigated by means of the rate equation approach. For the connection kernel A(k)=k^γ, the degree distribution n_k takes a power law if γ≥1 and decays stretched exponentially if 0≤γ< 1. We also consider a network system with the connection kernel A(k)=k^α(k_m-k)^β. It is found that n_k approaches a power law in the α> 1 case and has a stretched exponential decay in the 0≤α< 1 case, while it can take a power law with exponential truncation in the special α=β=1 case. Moreover, n_k may have a U-type structure if α> β.     

Anomalous luminescence of impurity-bound excitons in lithium borate crystals doped with cerium ions

The spectra and relaxation kinetics of the anomalous (?? < 10 ns) luminescence of Li₆GdB₃O₉:Ce³⁺ crystals have been experimentally detected. The time-resolved vacuum ultraviolet spectroscopy study has shown that optical transitions at 6.2 eV, caused by the transfer of an electron from the 4f ¹ ground state of Ce³⁺ to autoionizing states near the conduction band bottom of a crystal, lead to the formation of an impurity-bound exciton with the hole component localized on the 4f state of Ce³⁺ and the electron localized on states of the conduction band bottom. It has been found that the decay of such an exciton in Li₆GdB₃O₉:Ce³⁺ occurs through radiative recombination, leading to fast luminescence at 4.25 eV. The energy threshold for the formation of the impurity-bound exciton has been determined. The distribution functions of elementary relaxations over the reaction rate constants H(k), which determine the relaxation kinetics and luminescence quenching processes, have been calculated.     

Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

Two-dimensional random Lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap. In systems of finite spatial extension, the asymptotic decay of the survival probability is exponential and characterized by an escape rate γ, which can be related to the average positive Lyapunov exponent and to the dimension of the fractal repeller of the system. For infinite systems, the survival probability obeys a stretched exponential law of the form P(c,t)∼exp(−Ct^1/2). The transition between the two regimes is studied and we show that, for a given trap density, the non-integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space. Nevertheless, the repeller remains fractal. We determine the special scaling properties of this fractal.     

Luminescence of different modifications of crystalline silicon dioxide: Stishovite and coesite

Luminescence of very small samples of single crystals of coesite and stishovite has been studied. The spectra were detected under ionizing radiation (X-ray and electron beam) and the decay kinetics of cathodoluminescence in the range of time from 10 ns to 3 ms was measured. The coesite luminescence possesses a broad band at 3 eV with exponential decay about 680 μs at 80 K. The nature of this luminescence was explained as a self-trapped exciton creation in tetrahedron framework. The stishovite luminescence possesses two bands—blue (2.8 eV) and UV (4.7 eV). The UV band intensity grows more than 20 times with irradiation dose from initial level. This shows that the corresponding luminescence centers could be induced by the radiation. The decay of the UV band possesses a fast and a slow component. The determination of the fast decay parameters is beyond the capabilities of our apparatus (less than 10 ns), whereas the slow decay of the UV is non-exponential and takes place in the range of hundreds of microsecond. The blue band decay kinetics can be well approximated by power law ∼t⁻², which may correspond to recombination of defects created by radiation. The stishovite single crystal luminescence is very similar to that of germanium dioxide single crystal of rutile structure. The nature of the stishovite luminescence is explained as recombination of defects created by irradiation in octahedron-structured lattice.     

Dynamic hysteresis of magnetic aggregates with non-integer dimension

We investigate the dynamic hysteresis of nanoscale magnetic aggregates by employing Monte Carlo simulation, based on Ising model in non-integer dimensional space. The diffusion-limited aggregation (DLA) model with adjustable sticking probability is used to generate magnetic aggregates with different fractal dimension D. It is revealed that the exponential scaling law A(H₀, ω)∼H₀^α·ω^β, where A is the hysteresis area, H₀ and ω the amplitude and frequency of external magnetic field, applies to both the low-ω and high-ω regimes, while exponents α and β decrease with increasing D in the low-ω regime and keep invariant in the high-ω regime. A mean-field approach is developed to explain the simulated results.     

Dynamics and sedimentation velocity in charge-stabilized colloidal suspensions

Summary Charge-stabilized suspensions are characterized by the strong electrostatic interactions between the particles so that rather dilute systems may exhibit strong correlation resulting in a well-developed short-range order. This microstructure, quantitatively described by the pair distribution functiong(r), is rather different from that of (uncharged) hard spheres. It is shown how this difference affects the ?hydrodynamic function?H(k), which appears in the expression for the first cumulant Γ(k)=k ² D _eff(k)=k ² H(k)/S(k) of the dynamic autocorrelation function. Without hydrodynamic interaction,H(k)=D ⁰, which is the free-diffusion coefficient. Using pairwise additive hydrodynamic interaction and the lowest-order many-body theory of hydrodynamic interaction, it is found thatH(k) can deviate considerably fromD ⁰ even for systems of volume fractions ϕ as low as 10⁻³. These effects are more pronounced for collective diffusion than for self-diffusion. SinceH(k=0) is closely related to the sedimentation velocity, we have studied this quantity as a function of volume fraction. It is found that (H(0)/D ⁰) −1 scales asφ ^1/3 at low ϕ in salt-free suspensions. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Network topology and correlation features affiliated with European airline companies

The physics information of four specific airline flight networks in European Continent, namely the Austrian airline, the British airline, the France-Holland airline and the Lufthhansa airline, was quantitatively analyzed by the concepts of a complex network. It displays some features of small-world networks, namely a large clustering coefficient and small average shortest-path length for these specific airline networks. The degree distributions for the small degree branch reveal power law behavior with an exponent value of 2-3 for the Austrian and the British flight networks, and that of 1-2 for the France-Holland and the Lufthhansa airline flight networks. So the studied four airlines are sorted into two classes according to the topology structure. Similarly, the flight weight distributions show two kinds of different decay behavior with the flight weight: one for the Austrian and the British airlines and another for the France-Holland airline and the Lufthhansa airlines. In addition, the degree-degree correlation analysis shows that the network has disassortative behavior for all the value of degree k, and this phenomenon is different from the international airline network and US airline network. Analysis of the clustering coefficient (C(k)) versus k, indicates that the flight networks of the Austrian Airline and the British Airline reveal a hierarchical organization for all airports, however, the France-Holland Airline and the Lufthhansa Airline show a hierarchical organization mostly for larger airports. The correlation of node strength (S(k)) and degree is also analyzed, and a power-law fit S(k)∼k^1.1 can roughly fit all data of these four airline companies. Furthermore, we mention seasonal changes and holidays may cause the flight network to form a different topology. An example of the Austrian Airline during Christmas was studied and analyzed.     

Double-Exponentially Decayed Photoionization in CREI Effect： Numerical Experiment on 3D H2^＋

On the platform of the 3D H2^＋ system, we perform a numerical simulation of its photoionization rate under excitation of weak to intense laser intensities with varying pulse durations and wavelengths. A novel method is proposed for calculating the photoionization rate： a double exponential decay of ionization probability is best suited for fitting this rate. Confirmation of the well-documented charge-resonance-enhanced ionization （CREI） effect at medium laser intensity and finding of ionization saturation at high light intensity corroborate the robustness of the suggested double-exponential decay process. Surveying the spatial and temporal variations of electron wavefunctions uncovers a mechanism for the double-exponentially decayed photoionization probability as onset of electron ionization along extra degree of freedom. Henceforth, the new method makes clear the origins of peak features in photoionization rate versus internuclear separation. It is believed that this multi-exponentially decayed ionization mechanism is applicable to systems with more degrees of motion.     

The gamma decay of high spin states of39K

First will be exhibited, for the stationary case, a connection between the probability, to find a particle in the region of interaction, and the derivative of the scattering phase shift for the momentum. From the idea, that in stationary scattering a resonance is linked with an appreciable increase of this probability, one obtains new and quantitative criteria for the behavior ofδ _l(k). For instance, the nonresonant behavior can be characterised by the condition 2k dδ _l(k)/dk<1. The maximum of probability for the particle to be in the region of interaction, is considered in accordance with the criterium of maximal change of the phase shift, as a function ofk. This characterises the location of the resonance. Wigner's lower limit fordδ _l (k)/dk allows to formulate a resonance condition directly for the cross section. Furthermore will be discussed some connections with the pole approximations of theS-matrix and the “collision lifetime” of Smith.     

Growing networks with preferential deletion and addition of edges

A preferential attachment model for a growing network incorporating the deletion of edges is studied and the expected asymptotic degree distribution is analyzed. At each time step t=1,2,…, with probability π₁>0 a new vertex with one edge attached to it is added to the network and the edge is connected to an existing vertex chosen proportionally to its degree, with probability π₂ a vertex is chosen proportionally to its degree and an edge is added between this vertex and a randomly chosen other vertex, and with probability π₃=1−π₁−π₂<1/2 a vertex is chosen proportionally to its degree and a random edge of this vertex is deleted. The model is intended to capture a situation where high-degree vertices are more dynamic than low-degree vertices in the sense that their connections tend to be changing. A recursion formula is derived for the expected asymptotic fraction p_k of vertices with degree k, and solving this recursion reveals that, for π₃<1/3, we have p_k∼k^{−(3−7π₃)/(1−3π₃)}, while, for π₃>1/3, the fraction p_k decays exponentially at rate (π₁+π₂)/2π₃. There is hence a non-trivial upper bound for how much deletion the network can incorporate without losing the power-law behavior of the degree distribution. The analytical results are supported by simulations.     

Thermoluminescence and isothermal decay of luminescence in x-irradiated urea and thiourea

Experiments have been performed on the thermoluminescence and the isothermal decay of luminescence of both urea and thiourea crystals excited by X-rays at low temperature. The analysis of the isothermal decay curves has been performed assuming that the light emission consists of a sum of exponential decays with different time constants τ_i. Taking into account the dependence of τ_i from temperature, the values of thermal activation energies of the trapping centres involved in the luminescence process are obtained. In addition, the analysis of the glow curve, that has been performed through three different methods, allowed to obtain the values of some parameters related to the thermoluminescence process, such as the thermal activation energy E, the frequency factor s and the ration A/B between the retrapping and recombination probabilities. These results agree well enough with those from the isothermal decay analysis. A comparison has been made between the results for urea and for thiorea. Some topics concerned with the nature of the defects responsible for the luminescence process in these substances are also investigated.     

Spectroscopy and photophysics of chloro-bis-bipyridyl complexes of ruthenium(II) with pyridine ligands

The absorption, luminescence, and luminescence excitation spectra of ruthenium(II) complexes cis-[Ru(bpy)₂(L)Cl]⁺[bpy=2,2′-bipyridyl; L=NH₃, pyrazine, pyridine, 4-aminopyridine, 4-picoline, isonicotinamide, 4-cyanopyridine, 4,4′-bipyridyl, or trans-1,2-bis(4-pyridyl)ethylene] in alcoholic (4: 1 EtOH-MeOH) solutions are studied. At 77 K, the quantum yields and decay times of the luminescence of the complexes are measured and the deactivation rate constants of the lowest electronically excited metal-to-ligand charge transfer state (³MLCT) are determined. The linear correlation between the energy of the lowest state ³MLCT d _π(Ru)>π*(bpy) of the cis-[Ru(bpy)₂(L)Cl]⁺ complexes and the parameter pK_a of the free 4-substituted pyridines and pyrazine used as ligands is established.     

Unified index to quantifying heterogeneity of complex networks

Although recent studies have revealed that degree heterogeneity of a complex network has significant impact on the network performance and function, a unified definition of the heterogeneity of a network with any degree distribution is absent. In this paper, we define a heterogeneity index 0≤H<1 to quantify the degree heterogeneity of any given network. We analytically show the existence of an upper bound of H=0.5 for exponential networks, thus explain why exponential networks are homogeneous. On the other hand, we also analytically show that the heterogeneity index of an infinite power law network is between 1 and 0.5 if and only if its degree exponent is between 2 and 2.5. We further show that for any power law network with a degree exponent greater than 2.5, there always exists an exponential network such that both networks have the same heterogeneity index. This may help to explain why 2.5 is a critical degree exponent for some dynamic behaviors on power law networks.     

Probing spatial correlations in the inhomogeneous glassy state of the cuprates by Cu NMR

We discuss the crossover of the form of the Cu Nuclear magnetic resonance (NMR) spin echo decay at the onset of Cu wipeout in lanthanum cuprates. Experimentally, the echo decay undergoes a crossover from Gaussian to exponential form below the temperature where the Cu NMR intensity drops. The wipeout and the change in behavior both arise because the nuclei experience spatially inhomogeneous spin fluctuations at low temperatures. We argue that regions where the spin fluctuations remain fast are localized on length scales of order 1-2 lattice spacings. The inhomogeneity is characterized by the local activation energy E_a(r); we estimate the functional form of E_a(r) for points where E_a>(r)∼0.     

Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction

The finite size behavior of the susceptibility, Binder cumulant and some even moments of the magnetization of a fully finite O(n) cubic system of size L are analyzed and the corresponding scaling functions are derived within a field-theoretic ɛ-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as r ^{- (d + σ)}, where 2 < σ < 4, which does not change the short-range critical exponents of the system. Despite that the system belongs to the short-range universality class it is shown that above the bulk critical temperature T _c the finite-size corrections decay in a power-in-L, and not in an exponential-in-L law, which is normally believed to be a characteristic feature for such systems. Received 8 August 2001     

Monte Carlo study of the Ising antiferromagnetic with a longitudinal field on the anisotropic square lattice

The Ising antiferromagnetic in the presence of a magnetic field on an anisotropic square lattice is studied by Monte Carlo simulation. We obtained the phase diagram in the T-H plane investigating the reentrant behavior around of the critical field Hc=2Jy. Using the Binder cumulant we locate the critical temperature Tc as a function of H. In order to test our simulation, for null field we obtain the critical behavior of Tc as a function of r=Jy/Jx and is in excellent agreement with exact solution of Onsager. Our results indicate a second-order transition for all values of H and particular case r=1 (independent of the ratio r≠0), where not reentrant behavior was observed.     

A local-world node deleting evolving network model

A new type network growth rule which comprises node addition with the concept of local-world connectivity and node deleting is studied. A series of theoretical analysis and numerical simulation to the LWD network are conducted in this Letter. Firstly, the degree distribution p(k) of this network changes no longer pure scale free but truncates by an exponential tail and the truncation in p(k) increases as pa decreases. Secondly, the connectivity is tighter, as the local-world size M increases. Thirdly, the average path length L increases and the clustering coefficient 〈C〉 decreases as generally node deleting increases. Finally, 〈C〉 trends up when the local-world size M increases, so as to kmax. Hence, the expanding local-world can compensate the infection of the node deleting.     

Nearest-neighbor distances in microclusters

The average nearest-neighbor distances of model microclusters, which do not exhibit normal bulk-like packing of the atoms, have been calculated indirectly from the interference function, I(k), assuming that the packing of the microclusters is a normal lattice packing. Physically correct results obtained directly from the radial distribution function (RDF) differ from those calculated from I(k). The RDF nearest-neighbor distances generally exceed the bulk distances while those evaluated from I(k) are less than the bulk values. Recent experimental reports that the nearest-neighbor distances in microclusters, based on measurements of I(k), decrease with decreasing cluster size may thus be incorrect. The error arises from assuming the wrong packing in interpreting the interference function.