Bounds for the complex dielectric constant of a two-phase composite

We study the effective dielectric constant of a two-phase composite for the case where the dielectric constants of the two components are complex. Milton has derived a sequence of narrowing bounds in the complex plane limiting the possible values of the effective dielectric constant, assuming that certain geometrical coefficients characterizing the medium are known. We cast Milton's bounds in a different form, making use of auxiliary functions introduced by Bergman. We derive explicit expressions for the bounds up to third order and compare values for the corresponding geometrical coefficient derived for different geometries.     

On the Bergman-Milton bounds for the homogenization of dielectric composite materials

The Bergman-Milton bounds provide limits on the effective permittivity of a composite material comprising two isotropic dielectric materials. These provide tight bounds for composites arising from many conventional materials. We reconsider the Bergman-Milton bounds in light of the recent emergence of metamaterials, in which unconventional parameter regimes for relative permittivities are encountered. Specifically, it is demonstrated that: (a) for nondissipative materials the bounds may be unlimited if the constituent materials have relative permittivities of opposite signs; (b) for weakly dissipative materials characterized by relative permittivities with real parts of opposite signs, the bounds may be exceedingly large.     

Measurement of the complex dielectric constant of a single gold nanoparticle

A differential interference contrast microscopy technique that employs a photonic crystal fiber as a white-light source is used to measure both the real and the imaginary part of the complex dielectric constant of single 10 and 15 nm gold nanoparticles over a wavelength range of 480 to 610 nm. Noticeable deviations from bulk gold measurements are observed at short wavelengths and for individual particles even after taking into account finite-size surface damping effects.     

Rigorous lower and upper bounds for the slope parameter

It is proved within the framework of axiomatic field theory that the logarithmic derivative of the absorptive part of the scattering amplitude with respect to momentum transfer is bounded from above by $\text{(15}\text{log}\text{s)}\text{[4√t(2 ? √t)]}$ for a sequence of s→+∞, and from below either in the s-channel by const. × s^?5 log^?4s or in the u-channel by const. × u^?5 log^?4u for at least one sequence of s or u →+∞, respectively. In the particular case of the s?u even-symmetric amplitude, a stronger lower bound is obtained; namely, const. × s^?5 log^?4s for at least one sequence of s→+∞. Here s, t, and u are the usual Mandelstam variables, and all bounds are obtained in the forward and the unphysical regions: 0?t<4 (in units of pion mass).It is observed that the Regge amplitude β(t)s^α(t) of high-energy scattering gives the same energy dependence as the above upper bound, and, furthermore, that the slope of the Regge trajectory is bounded from above by $\text{15}\text{[4√t(2 ? √t)]}\text{for}\text{0 < t < 4}$.     

Rigorous bounds on quark masses within a nonrelativistic approach

By applying the Feynman-Hellmann theorem to \(q\bar q\) systems we find the following bounds on quark mass differences from the spectrum ofall quarkonium states $$\begin{gathered} 0.27 \leqq m_s - m_u \leqq 0.45GeV \hfill \\ 1.23 \leqq m_c - m_s \leqq 1.46GeV \hfill \\ 3.30 \leqq m_b - m_c \leqq 3.55GeV. \hfill \\ \end{gathered}$$ As best values we derive $$\begin{gathered} m_u = m_d = 0.31GeV,m_s = 0.62GeV, \hfill \\ m_c = 1.91GeV,m_b = 5.27GeV. \hfill \\ \end{gathered}$$      

Quasi-optical measurement of complex dielectric constant at 300 GHz

A two beam interferometer in the Martin-Puplett configuration is used to determine the complex dielectric constant at 300 GHz of teflon, TPX-plastics, SPECTRALON and paraffin waxes with melting temperatures of 48° C and 72° C, respectively. The design of the quasi-optical system leads to a constant beam diameter at the power detector independent of path delay and frequency. The power detector signal is recorded not only along one period but over about 50 periods. A spectrum estimation routine allows to determine more exactly amplitude and phase angle of the signal. A basic problem is noticed: imperfect detector and source match cause harmonic distortion of the power detector signal. The effects on processing the loss tangent and the invalidation are shown. Finally loss tangent and dielectric constant are determined indirectly by optimizing an equivalent microwave circuit using a commercial available microwave design system to take multiple reflections and losses in consideration.     

The dielectric constant of a composite material—A problem in classical physics

The problem of calculating the effective dielectric constant of a composite material ε_e is analyzed by separating out the dependence of ε_e on the microscopic geometry. A set of characteristic geometric functions is defined, which depend in detail on the microscopic geometry of the material under discussion, but whose general analytical properties can be derived. The pole structure of these functions is shown to have important consequences. It is experimentally observable in certain optical and microwave experiments on metal-insulator composites. Furthermore, it can be used to obtain a considerable amount of quantitative information about ε_e in the form of rigorous upper and lower bounds. Finally, it allows a new approach to be taken in the discussion of the critical properties of composite systems near a percolation threshold.The characteristic functions defined and discussed in this article are applicable not only to ε_e, but also to other scalar material constants of the composite material, such as the magnetic permeability, electrical and thermal conductivity, and diffusivity.     

Semi-phenomenological bounds for the pion-pion coupling constant

Upper and lower bounds on the Chew-Mandelstam ππ coupling constant λ are derived as functions of the D-wave scattering length a₂. The only information used is the axiomatic analyticity domain of the ππ scattering amplitude, and the crossing and unitarity properties. For a₂ = 7 × 10^?4, one obtains ?0.164<λ<0.162. This is compared with previous bounds.     

Cluster expansion for the dielectric constant of a polarizable suspension

We derive a cluster expansion for the electric susceptibility kernel of a dielectric suspension of spherically symmetric inclusions in a uniform background. This also leads to a cluster expansion for the effective dielectric constant. It is shown that the cluster integrals of any order are absolutely convergent, so that the dielectric constant is well defined and independent of the shape of the sample in the limit of a large system. We compare with virial expansions derived earlier in statistical mechanics for the dielectric constant of a nonpolar gas. In these expansions the virial coefficients are given by integrals which are only conditionally convergent.     

Rigorous solution of a problem for a cylindrical cavity containing a ferrite and a dielectric

A rigorous solution is given for modes of TM_nmo type. The complex resonant frequency _n is expressed via a transcendental characteristic equation. Some particular cases are discussed.Read at the Third All-Union Conference on Ferrites, Leningrad, 23 October 1963.     

On the dielectric constant for a non-polar fluid,a comparison

Sullivan and Deutch recently showed how the Wertheim formula for the dielectric constant of a non-polar fluid can be obtained as the lowest order result in a systematic expansion. We compare the results of this procedure with the results obtained with the more usual analysis of deviations from Clausius-Mossotti. On the basis of some rigorous theoretical arguments and some numerical results we conclude that Wertheim's formula predicts deviations from Clausius-Mossotti accurately to second order in the density. For higher densities we find that higher order corrections in the systematic expansion are necessary to find agreement. It is concluded that Wertheim's formula is therefore only a valid improvement over Clausius-Mossotti for low and intermediate densities. Sullivan and Deutch derive an expression for the local field factor, which appears in light scattering, valid to lowest order. They compare this factor with experimental results and find that it agrees much better than the factor which follows from Clausius-Mossotti for liquid densities. We give a general expression and show that higher order corrections to Wertheim's theory change the local field factor by a few percent. On the basis of this we conclude that the better than one percent agreement found by Sullivan and Deutch is to some extent fortuitous.     

Rigorous bounds on the storage capacity of the dilute Hopfield model

We study a neural network model consisting ofN neurons where a dendritic connection between each pair of neurons exists with probabilityp and is absent with probability 1-p. For the Hopfield Hamiltonian on such a network, we prove that ifp c[(lnN)/N]^1/2, the model can store at leastm= _cpN patterns, where _c 0.027 ifc 3 and decreases proportional to 1/(–lnc) forc small. This generalizes the results of Newman for the standard Hopfield model.     

Rigorous absolute bounds for pion-pion scattering (1). Solving a first extremum problem

This paper is the first of a series of three where we reconsider the derivation of rigorous absolute bounds for strong interactions. It is devoted to the solution of a preliminary extremum problem. From the knowledge, at a given physical energy s and unphysical angle cos θ₀ > 1, of the absorptive part A (s, cos θ₀) of the elastic scattering amplitude for two spinless particles, we find, by exploiting the unitarity condition, the least upper bound of the modulus |F(s, cos θ₁)| of the scattering amplitude for the same energy s and various angles θ₁. Upper (but not least upper) bounds given by previous authors are numerically compared with ours.     

Longitudinal dielectric constant for quantum wells

We have calculated the longitudinal dielectric constant for a GaAs-AlGaAs quantum well structure. In particular, the total contribution to the transitions is taken as a sum of separate contributions of the regions around the Λ, X, and L symmetry points of the Brillouin zone. Since the electron- and hole-state are quantized in a well formed by potential barriers of the hetrojunction, the static dielectric constant is significantly reduced. For a well width of 2 nm and barriers formed by Al_.35Ga_.65As, the reduction is almost 20%.