Analytic structure and explicit solution of an important implicit equation

The equationz=2G(z)?expG(z)+1 (and similar ones obtained from it by substitutions) appears in connection with a variety of problems ranging from pure mathematics (combinatorics; some first order, nonlinear differential equations) over statistical thermodynamics to renormalization theory. It is therefore of interest to solve this equation forG(z) explicitly. It turns out, after study of the complex structure of thez andG planes, that an explicit integral representation ofG(z) can be given, which may be directly used for numerical calculations of high precision.     

A comparison between radiative and non-radiative deexcitation after electron capture by multiple charged ions

For 100 ke V Ne^z+ (z = 1?4) and 200 keV Ar^z+ (z = 1?8) colliding with noble gas atoms the fraction of electron capture processes followed by radiationless deexcitation is shown to increase with rise of z and of target atomic number.     

Low temperature sintering of sub-stoichiometric Ni-Cu-Zn ferrites: Shrinkage, microstructure and permeability

We have studied sub-stoichiometric Ni-Cu-Zn ferrites with iron deficiency (i.e., <50mol% Fe₂O₃) of composition Ni_0.20Cu_0.20Zn_0.60+zFe_2−zO_4−(z/2) with 0≤z≤0.06. The temperature of maximum shrinkage rate is shifted from T=1000 °C for z=0 towards lower temperatures down to T=900 °C for a sub-stoichiometric ferrite with z=0.02. Dense samples are obtained after firing at 900 °C for z>0 only. Sub-stoichiometric compositions (z>0) do not form single-phase spinel ferrites after sintering at 900 °C, but rather represent mixtures of CuO and a stoichiometric ferrite with slightly modified composition. The formation of small amounts of CuO at grain boundaries is demonstrated by XRD and SEM. The permeability is increased from μ=80 for stoichiometric ferrites (z=0) to μ=660 for z=0.02. The formation of CuO during sintering of sub-stoichiometric ferrites supports densification and is a prerequisite for low temperature firing of multilayer inductors. Addition of 1 wt% Bi₂O₃ as liquid phase sintering aid is required to provide sufficient densification of the stoichiometric ferrite (z=0) at 900 °C. Addition of 0.37 wt% Bi₂O₃ to a sub-stoichiometric ferrite (z=0.02) results in dense samples after firing at 900 °C; however, the microstructure formation is dominated by heterogeneous grain growth.     

On a soluble model of an antiferromagnetic chain with Dzyaloshinsky interactions. II

Time-dependent properties of the one-dimensional XY model with Dzyaloshinsky interactions in the presence of a magnetic field in the z direction are investigated. Explicit expressions are derived for the time correlation functions of the z components of two spins and the time auto-correlation functions of M^z (= the z component of the magnetization). The ergodic behaviour of M^z in the thermodynamic limit is discussed in some detail. Furthermore an exact expression is derived for the time dependence of the average of M^z in a typical nonequilibrium situation. Finally the frequency-dependent susceptibility is evaluated.     

Application of the two-mode squeezed coherent state representation in deriving generalized optical Collins formula

We propose a new two-mode squeezed coherent state representation |z₁, z₂〉_g which is characteristic of the correlation between the squeezing and the displacement. Based on it and using the technique of integration within an ordered product of operators we obtain a generalized two-mode Fresnel operator (GTFO), which is an image of the mapping from (z₁, z₂) to in |z₁, z₂〉_g representation. The matrix element of GTFO in the coordinate representation leads to a generalized two-dimensional Collins formula (Huygens-Fresnel integration transformation describing optical diffraction) in entangled form.     

Nonlinear evolution of the axisymmetric nuclear surface

We consider a uniformly charged incompressible nuclear liquid bounded by a closed surface. It is shown that the evolution of an axisymmetric surface Г(r, t) ≡ σ ? ∑(z, t) = 0, r = (σ, φ, z) can be approximately reduced to the motion of a curve in the (σ, z) plane. A nonlinear integro-diffrerential equation for the contour Σ (z, t) is derived. The contour Σ (z, t) and the local curvature are found to be a direct correspondence, which makes it possible to use methods of differential geometry to analyze the evolution of an axisymmetric nuclear surface.     

Forced ferromagnetic ordering in systems with magnetic ions of two types

The distribution of magnetization in domain walls of ferromagnetic (FM) crystals with two order parameters interacting with each other—F_1z and F_2z, is considered. In one case the interaction is represented as (F_1zF_2z) and in another—(F_1z²F_2z²). It is shown that in the former case the magnetization distribution has space amplitude modulation, and in latter case appears space frequency modulation. Changes of the paramagnetic-ferromagnetic phase transition temperature in view of interaction of two FM subsystems have been studied.     

Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence

The analytical expressions for the effective Rayleigh range z_R of Gaussian array beams in turbulence for both coherent and incoherent combinations are derived. It is shown that z_R of Gaussian array beams propagating through atmospheric turbulence depends on the strength of turbulence, the array beam parameters and the type of beam combination. For the coherent combination z_R decreases due to turbulence. However, for the incoherent combination there exists a maximum of z_R as the strength of turbulence varies. The z_R of coherently combined Gaussian array beams is larger than that of incoherently combined Gaussian array beams, but for the coherent combination case z_R is more sensitive to turbulence than that for the incoherent combination case. The larger the beam number is, the longer z_R is, and the more z_R is affected by turbulence. For the coherent combination z_R is not monotonic versus the relative beam separation distance, and the effect of turbulence on z_R is appreciable within a certain range of the relative beam separation distance.     

The elliptical dimension of space-time atmospheric stratification of passive admixtures using lidar data

State-of-the-art airborne lidar data of passive scalars have shown that the spatial stratification of the atmosphere is scaling: the vertical extent (Δz) of structures is typically ≈Δx^H_z where Δx is the horizontal extent and H_z is a stratification exponent. Assuming horizontal isotropy, the volumes of the structures therefore vary as ΔxΔxΔx^H_z=Δx^D_s where the “elliptical dimension” D_s characterizes the rate at which the volumes of typical non-intermittent structures vary with scale. Work on vertical cross-sections has shown that 2+H_z=2.55±0.02 (close to the theoretical prediction 23/9).In this paper we extend these (x, z) analyses to (z, t). In the absence of overall advection, the lifetime Δt of a structure of size Δx varies as Δx^H_t with H_t=2/3 so that the overall space-time dimension is D_st=29/9=3.22…. However, horizontal and vertical advection lead to new exponents: we argue that the temporal stratification exponent H_t≈1 or ≈0.7 depending on the relative importance of horizontal versus vertical advection velocities. We empirically test these space-time predictions using vertical-time (z, t) cross-sections using passive scalar surrogates (aerosol backscatter ratios from lidar) at ∼3 m resolution in the vertical, 0.5-30 s in time and spanning 3-4 orders of magnitude in scale as well as new analyses of vertical (x, z) cross-sections (spanning over 3 orders of magnitude in both x, z directions). In order to test the theory for density fluctuations at arbitrary displacements in (Δz, Δt) and (Δx, Δz) spaces, we developed and applied a new Anisotropic Scaling Analysis Technique (ASAT) based on nonlinear coordinate transformations. Applying this and other analyses to data spanning more than 3 orders of magnitude of space-time scales we determined the anisotropic scaling of space-time finding the empirical value D_st=3.13±0.16. The analyses also show that both cirrus clouds and aerosols had very similar space-time scaling properties. We point out that this model is compatible with (nonlinear) “turbulence” waves, hence potentially explaining the observed atmospheric structures.     

Fractal Dimension of Julia Set for Nonanalytic Maps

The Hausdorff dimensions of the Julia sets for nonanalytic maps f(z) = z ²+εz* and f(z)=z*²+ε are calculated perturbatively for small ε. It is shown that Ruelle's formula for the Hausdorff dimensions of analytic maps cannot be generalized to nonanalytic maps.     

Evidence for higher twist effects in fast π− production by antineutrinos in neon

Evidence for a significant higher twist contribution to highz π^? production in antineutrino scattering is presented. In events withW>3 GeV andQ ²>1 GeV² in our data, it accounts for (51 ±8)% of all π^? withz above 0.5. It is consistent with thez?Q ² correlations of Berger's higher twist prediction. The data are inconclusive concerning the predictedy?z correlation andp _T dependence. Thez ?Q ² correlation is not adequately described by the Lund Monte-Carlo.     

Tip-enhanced Raman scattering: Influence of the tip-surface geometry on optical resonance and enhancement

The tip-sample distance (z) dependence of tip-enhanced Raman scattering (TERS) has been investigated. The intensities of both, the Raman lines and the broad TERS background, exhibit strong decays with increasing z, which are nearly complete within 10 nm withdrawal of the STM tip in z direction. Interestingly, the maximum of the broad Lorentzian-shaped TER background is substantially blue shifted in energy with z. This effect is ascribed to a corresponding blue shift of the energies of localized plasmon modes upon tip retraction. Both experimental results fit very well data of a simple theoretical near-field model.     

Anisotropy profiles of optical fibers and preforms

An experimental method to measure the optical anisotropy profiles of optical fibers and preforms is presented. The local values of n_z?n_r and n_z?n_?, (r,?,z) being a cylindrical coordinate system associated with the fiber or preform, are obtained.     

Dynamical connections between quark fragmentation functions

Assuming that the quark kicked by a virtual photon emits pions one by one, integral equations connecting the fragmentation functions D_u^π+ (z) and D_u^π? (z) are obtained. It is shown that they have a plateau the height of which can be determined from the multiplicity difference N_p^π+ (z) ? N_p^π? (z). Comparison with experiment is made.     

The application of the differential operator method to the Blume-Emery-Griffiths model

Within an effective-field approximation, general expressions for evaluating the second-order phase transition line and the tricritical point of the anisotropic Blume-Emery-Griffiths model are obtained by the use of the differential operator technique. The phase diagrams and the behavior of the tricritical point are investigated numerically for the honeycomb lattice (z = 3) and square lattice (z = 4). We find a new disordered phase which may correspond to the staggered quadrupolar phase predicted in the Monte Carlo simulation, when some conditions are satisfied. The phase diagrams for z = 3 and z = 4 systems exhibit a reentrant behavior for positive values of the uniaxial anisotropy parameter. The change of the tricritical point with the value of the reduced biquadratic parameter is also studied for the system with z = 3.     

Spin-glass-like behavior in ZnxCryAlzSe4

The magnetic and electrical properties of the Al-doped polycrystalline spinels Zn_xCr_yAl_zSe₄ (0.13≤z≤0.55) with the antiferromagnetic (AFM) order and semiconducting behavior were investigated. A complex antiferromagnetic structure below a Néel temperature T_N≈23 K for the samples with z up to 0.4 contrasting with the strong ferromagnetic (FM) interactions evidenced by a large positive Curie-Weiss temperature θ_CW decreasing from 62.2 K for z=0.13 to 37.5 K for z=0.55 was observed. Detailed investigations revealed a divergence between the zero-field-cooling (ZFC) and field-cooling (FC) susceptibilities at temperature less than T_N suggesting bond frustration due to competing ferromagnetic and antiferromagnetic exchange interactions in the compositional range 0.13≤z≤0.4. Meanwhile, for z=0.55 a spin-glass-like behavior of cluster type with randomly oriented magnetic moments is observed as the ZFC-FC splitting goes up to the freezing temperature T_f=11.5 K and the critical fields connected both with a transformation of the antiferromagnetic spin spiral via conical magnetic structure into ferromagnetic phase disappear.     

The proton recoil spectrum of thermal neutrons decaying in vacuum

The decay of neutrons in an evacuated tangential reactor beam tube is considered. Due to the thermal motion of the decaying neutrons, the proton recoil spectrum in the laboratory system (z-axis parallel to the beam tube) differs from that in the c.m. system. This difference is calculated for a point at the beam tube exit, and expressed in terms of the average¯q _z and of the variance of thez-component of neutron momentum. An activation method for measuring the neutron current densityJ _z (which is closely related to¯q _z) is suggested. The distortion of the spectrum by the finite resolution of an electrostatic spectrometer is also given. The results are to be used in connection with a planned measurement of the proton recoil spectrum in neutron decay.     

On the scaling factors α(z) and δ(z)

We introduce a new method to determine the scaling factors α(z) and δ(z) for the period-doubling route to chaos in dissipative systems, exemplified by the one-dimensional mapping x_n+1=1−λ∣x_n∣^z. With the help of the Feigenbaum universal functions g(x) and h(x) we derive the inequality α^z−α<δ(z)<α^z, implying in particular that δ(z) remains finite (≲30) in the limit z → ∞.     

Overdamped motion of interacting particles in general confining potentials: time-dependent and stationary-state analyses

By comparing numerical and analytical results, it is shown that a system of interacting particles under overdamped motion is very well described by a nonlinear Fokker-Planck equation, which can be associated with nonextensive statistical mechanics. The particle-particle interactions considered are repulsive, motivated by three different physical situations: (i) modified Bessel function, commonly used in vortex-vortex interactions, relevant for the flux-front penetration in disordered type-II superconductors; (ii) Yukawa-like forces, useful for charged particles in plasma, or colloidal suspensions; (iii) derived from a Gaussian potential, common in complex fluids, like polymer chains dispersed in a solvent. Moreover, the system is subjected to a general confining potential, ??(x)?=?(??|x| ^z )/z (???>?0, z?>?1), so that a stationary state is reached after a sufficiently long time. Recent numerical and analytical investigations, considering interactions of type (i) and a harmonic confining potential (z?=?2), have shown strong evidence that a q-Gaussian distribution, P(x,t), with q?=?0, describes appropriately the particle positions during their time evolution, as well as in their stationary state. Herein we reinforce further the connection with nonextensive statistical mechanics, by presenting numerical evidence showing that: (a) in the case z?=?2, different particle-particle interactions only modify the diffusion parameter D of the nonlinear Fokker-Planck equation; (b) for z????2, all cases investigated fit well the analytical stationary solution P _st(x), given in terms of a q-exponential (with the same index q?=?0) of the general external potential ??(x). In this later case, we propose an approximate time-dependent P(x,t) (not known analytically for z????2), which is in very good agreement with the simulations for a large range of times, including the approach to the stationary state. The present work suggests that a wide variety of physical phenomena, characterized by repulsive interacting particles under overdamped motion, present a universal behavior, in the sense that all of them are associated with the same entropic form and nonlinear Fokker-Planck equation.     

Spinel block doping and conductivity of sodium beta″-alumina ceramics

Sodium ion conducting fully stabilized, single phase β″-Al₂O₃ ceramics of compositions Na_1+z+2z'M_zLi_z'Al_11?z?z'O[in17, M=Mg or Ni, have been investigated. The conductivities of these ceramics do not depend upon the kind of stabilizing ions introduced into the spinel blocks, but predominantly depend upon the composition of the conduction slab. For fully magnesium stabilized ceramics with M=Mg and z′= 0, conductivity at 300°C is maximal for 3z = 2.0…2.2.