Photoelectron Energy Loss in Al(002) Revisited: Retrieval of the Single Plasmon Loss Energy Distribution by a Fourier Transform Method

A Fourier transform (FT) algorithm is proposed to retrieve the energy loss function (ELF) of solid surfaces from experimental X-ray photoelectron spectra. The intensity measured over a broad energy range towards lower kinetic energies results from convolution of four spectral distributions: photoemission line shape, multiple plasmon loss probability, X-ray source line structure and Gaussian broadening of the photoelectron analyzer. The FT of the measured XPS spectrum, including the zero-loss peak and all inelastic scattering mechanisms, being a mathematical function of the respective FT of X-ray source, photoemission line shape, multiple plasmon loss function, and Gaussian broadening of the photoelectron analyzer, the proposed algorithm gives straightforward access to the bulk ELF and effective dielectric function of the solid, assuming identical ELF for intrinsic and extrinsic plasmon excitations. This method is applied to aluminum single crystal Al(002) where the photoemission line shape has been computed accurately beyond the Doniach–Sunjic approximation using the Mahan–Wertheim–Citrin approach which takes into account the density of states near the Fermi level; the only adjustable parameters are the singularity index and the broadening energy Г (inverse hole lifetime). After correction for surface plasmon excitations, the q-averaged bulk loss function, <Im[??1 / ε(E, q)]> _q , of Al(002) differs from the optical value Im[??1 / ε(E, q?=?0)] and is well described by the Lindhard–Mermin dispersion relation. A quality criterion of the inversion algorithm is given by the capability of observing weak interband transitions close to the zero-loss peak, namely at 0.65 and 1.65 eV in ε(E, q) as found in optical spectra and ab initio calculations of aluminum.     

Thickness dependence of the surface plasmon dispersion in ultrathin aluminum films on silicon

The collective excitation in Al films deposited on Si(1 1 1)-7 × 7 surface was investigated by high-resolution electron-energy-loss spectroscopy (HREELS), X-ray photoelectron spectroscopy (XPS) and scanning tunneling microscopy (STM). At the Al film thickness d < 10 ML, the surface plasmon of Al film has only a small contribution to the observed energy-loss peaks in the long wavelength limit (q_∥≈0), while its contribution becomes significant for q_∥>d^-1. More interestingly, for thin Al films, the initial slope of the surface plasmon dispersion curve is positive at q_∥∼0, in a sharp contrast to bulk Al surface where the energy dispersion is negative. These observations may be explained based on the screening interaction of the space charge region at the Al-Si interface.     

Radiative plasmon polaritons in multilayer structures with a two-dimensional electron gas

Two-dimensional plasmon polaritons are analyzed for a typical experimental configuration in which a layer of two-dimensional electrons with a finite mobility lies on the top of a dielectric waveguide formed by the substrate (a wafer of finite thickness). Two-dimensional plasmons couple strongly to the radiative modes of this dielectric waveguide. It is shown that, as a result of the competition between collisional and radiative processes, a family of eight quasi-stationary normal modes arises. Six of them decay carrying energy to infinity. The two remaining plasmon-polariton modes are nonradiative. One of these modes, the TM-type plasmon polariton, in the limiting case where retardation is disregarded corresponds to the conventional longitudinal two-dimensional plasmon. The other mode, the TE-type plasmon polariton, exists only for a finite thickness of the substrate. All of them are characterized by different dispersion relations of the complex frequency ω(q) = Reω + iImω and differ in both polarization (longitudinal and transverse) and symmetry with respect to the direction of decay (symmetric and asymmetric). The latter modes decay slowly, propagating into free space to plus or minus infinity. The conditions under which the Q factors of certain modes are arbitrarily high are found. In this case, Imω(q ₀) = 0, and dissipative losses in the two-dimensional electron gas are compensated by external sources. As a result, the reflection coefficient for a plane wave whose angle of incidence is determined by the vector q ₀ vanishes.     

Plasmon hybridization in metallic nanotubes

We study theoretical formalism of the plasmon hybridization in a metallic nanotube and find an explicit form of the dispersion relation for surface plasmons, in terms of interaction between the bare plasmon modes of the individual surfaces of the nanotubes. In the special case when the longitudinal wave vector is zero (q=0), the plasmon hybridization of a nanotube has a behavior similar to the spherical nanoshell.     

Plasmon dispersion and intersubband resonance at high wavevectors in Si(100) inversion layers

Deviations from the classical two-dimensional plasmon dispersion are observed at high wavevectors q in electron inversion layers on Si(100) MOS-capacitors with periodically structured gate electrodes. For high inversion electron densities (n_s> 6 × 10¹²cm^?2) an unexpected mass enhancement is extracted from the plasmon dispersion. In addition the plasmon linewidth is found to be significantly larger than predicted from transport experiments. On the same samples non-vertical intersubband resonance transitions are observed with radiation incident normal to the interface. They are excited by an electric field component normal to the surface that is induced by the structured gate electrode.     

Dispersion relation for the 3.8 eV volume plasmon of silver

The dispersion curve of the 3.8 eV volume plasmon of Ag has been obtained by electron energy loss measurements for wave vectors q in the range $\text{0.1 ?q? 0.35}\text{A}\text{?}{}^{-1}$. Results from the low-q end of this range match well with the optical results of Lindau and Nilsson. The curve can be described by a dispersion constant a = 0.8 ± 0.1 and the intensity of the loss goes like I≈q^-2.9±0.4. A simple model for the longitudinal dielectric function yields results in good agreement with the experimental conclust     

Anisotropy of the 15 eV volume plasmon dispersion in A1

A dependence of plasmon dispersion for epitaxial aluminum films has been found for the directions [100], [110] and [111]. A differene of about 0.5 eV has been observed between the directions [100] and [110] at q = 1.4 Å^?1. This effect is due to the influence of lattice on the electron gas.     

On the Theory of Plasmon Dispersion in Electron-Doped Cuprates

An explicit expression for the dynamic charge susceptibility for electron-doped cuprates has been derived. This expression accurately reproduces the wave vector dependence of the plasmon frequency observed in inelastic X-ray scattering experiments for Nd_{2 – x}Ce_xCuO₄. The imaginary part of the charge susceptibility along the triangular path in the Brillouin zone is plotted. It is demonstrated that the spectral weight of the plasmon mode near q = 0 is negligibly low. The calculated frequencies of the plasmon mode for all wave vectors in the Brillouin zone turn out to lie outside the range of damping related to electron?hole excitations. A formula for the charge susceptibility is derived within the t?t′?t″?J model supplemented by the Coulomb interaction operator and three-site terms. The derivation is performed by the Green’s function technique employing the formalism of composite Hubbard operators and the Mori projection method, which have proved themselves in the analysis of collective spin excitations. The used Fourier transform of the Coulomb interaction corresponds to the monolayer model with a spatially periodic structure, which is embedded in a three-dimensional crystal lattice.     

Tsallis q-Stat and the Evidence of Long-Range Interactions in Soil Temperature Dynamics

The complexities in the variations of soil temperature and thermal diffusion poses a physical problem that requires more understanding. The quest for a better understanding of the complexities of soil temperature variation has prompted the study of the q-statistics in the soil temperature variation with the view of understanding the underlying dynamics of the temperature variation and thermal diffusivity of the soil. In this work, the values of Tsallis stationary state q index known as q-stat were computed from soil temperature measured at different stations in Nigeria. The intrinsic variations of the soil temperature were derived from the soil temperature time series by detrending method to extract the influences of other types of variations from the atmosphere. The detrended soil temperature data sets were further analysed to fit the q-Gaussian model. Our results show that our datasets fit into the Tsallis Gaussian distributions with lower values of q-stat during rainy season and around the wet soil regions of Nigeria and the values of q-stat obtained for monthly data sets were mostly in the range 1.2≤q≤2.9 for all stations, with very few values q closer to 1.2 for a few stations in the wet season. The distributions obtained from the detrended soil temperature data were mostly found to belong to the class of asymmetric q-Gaussians. The ability of the soil temperature data sets to fit into q-Gaussians might be due and the non-extensive statistical nature of the system and (or) consequently due to the presence of superstatistics. The possible mechanisms responsible this behaviour was further discussed.     

Many-body effects in the binding energies of image states at surfaces

A study of the influence of many-body effects on binding energies and effective masses of image potential induced surface states is presented. The binding energies calculated using experimentally measured ε(ω) response functions show a dependence on the electron density different from that obtained using a plasmon model, and are in agreement with recent experimental data of diesen et al. [Phys. Rev. B33 (1986) 5241]. The effects of plasmon dispersion and single particle excitations on binding energies are evaluated with a simple model, and it is shown that they result in a strong cancellation in the range of r_s values where most experimental data have been obtained (r_s ≈ 3), if realistic values of the surface plasmon dispersion relation are used. Corrections to the effective masses due to many-body effects are shown to be of the order of 1%.     

Electromagnetic form factors for nucleons and pions at positive and negative q 2 in the model of quark-gluon strings

The electromagnetic form factors for pions and nucleons are considered within the model of quark-gluon strings, where the momentum-transfer dependence of hadronic form factors is determined by the intercepts of the corresponding Regge trajectories and by the Sudakov form factor. Analytic expressions found for form factors in the timelike region admit an analytic continuation to the spacelike region. The resulting form factors for pions and nucleons comply well with experimental data both for positive and for negative values of the squared momentum transfer q ². It is shown that the distinctions between the absolute values of the pion and nucleon form factors F _π(q ²), G _m (q ²), and F ₂(q ²) at positive values of q ² and those at negative values of this variable are associated with the analytic properties of the double-logarithmic term in the exponent of the Sudakov form factor. The spin structure of the amplitudes for quark transitions into hadrons that is proposed in the present study makes it possible to describe fairly well available experimental data on the Pauli form factor F ₂ and on the ratio G _e /G _m .     

Some observations on Levy stability and intermittency in nucleus-nucleus interactions at SPS energies

The power law relation between higher order and second order scaled factorial moments is studied in one dimensional pseudo-rapidity phase (η) space in the interactions of ³²S beam with CNO, AgBr and Emulsion at incident energy of 200 AGeV. Observation for such a power law may indicate a self similar cascade mechanism in multiparticle production process. The values of slope, β_q are found to be independent of target size. The value of the scaling exponent υ = 1.412 obtained is higher than the critical value υ = 1.304, indicating that no second order phase transition exists in our data. The ratio of anomalous fractal dimensions, d_q/d₂ is found to increase with increase in the order of moments, q. The dependence of d_q/d₂ on q indicates a multifractal structure and the presence of self-similar cascading mechanism in our data. The d_q/d₂ values are well described by the Levy-stable distribution with Levy index μ = 1.562 which is consistent with and lies within the Levy stable region (0 ≤ μ ≤ 2). The multifractal spectrum is concave downward with a maximum at q = 0. The decrease in D_q with increasing q shows that there is a self affine multifractal behaviour in multiparticle production in our data.     

On the scale dependence of the dimension of hadronic matter

We advance the notion of a dynamical, scale-dependent Hausdorff dimension for hadronic matter. The physical motivations behind such a concept are explained. For very small scales, we relate this dimension to the anomalous dimension of the product of two momentum densities. For hadrons, we rely on the precocious onset of Bjorken scaling in e-p inelastic scattering and a heuristic parton model to obtain the Hausdorff dimensional profile for the proton d(q²) as a function of the probing photon momentum q². We find that d reaches a maximum around d = 1 for a range of low q² values. This feature is indicative of a string-like structure of the hadrons at low momentum transfers.     

The maximization of Tsallis entropy with complete deformed functions and the problem of constraints

By only requiring the q deformed logarithms (q exponentials) to possess arguments chosen from the entire set of positive real numbers (all real numbers), we show that the q-logarithm (q exponential) can be written in such a way that its argument varies between 0 and 1 (among negative real numbers) for 1?q<2, while the interval 0<q?1 corresponds to any real argument greater than 1 (positive real numbers). These two distinct intervals of the nonextensivity index q, also the expressions of the deformed functions associated with them, are related to one another through the relation (2−q), which is so far used to obtain the ordinary stationary distributions from the corresponding escort distributions, and vice versa in an almost ad hoc manner. This shows that the escort distributions are only a means of extending the interval of validity of the deformed functions to the one of ordinary, undeformed ones. Moreover, we show that, since the Tsallis entropy is written in terms of the q-logarithm and its argument, being the inverse of microstate probabilities, takes values equal to or greater than 1, the resulting stationary solution is uniquely described by the one obtained from the ordinary constraint. Finally, we observe that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in (−∞,0]. However, this case corresponds to, although related, a different entropy expression than the Tsallis entropy.     

Theory of lattice softening and spin-susceptibility in A15-compounds

The one-dimensional model of A15-compounds is used to calculate phonon frequencies Ω_λ (q) from temperature dependent screening properties of the electron system. The interrelationship between the magnetic susceptibility χ(q) and Ω_λ (q) is derived and both quantities are studied in the longwavelength limit ¦q¦=0 and for ¦q¦=2 ·k _itF. Numerical values are obtained for V₃Si and Nb₃Sn.     

Momentum distributions in the nucleus

The nuclear form factor F(q) and one particle momentum distribution p(q) can be shown to have a power law decrease for large momenta. For the form factor F(q) we show that it is q/A that must be large for this asymptotic behavior to be important. For only q large the form factor, in a simple model, is shown to decrease exponentially in q. A similar behavior for p(q) is proposed.     

Renormalisation of q-Regularised Multiple Zeta Values

We consider a particular one-parameter family of q-analogues of multiple zeta values. The intrinsic q-regularisation permits an extension of these q-multiple zeta values to negative integers. Renormalised multiple zeta values satisfying the quasi-shuffle product are obtained using an Hopf-algebraic Birkhoff factorisation together with minimal subtraction.     

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation on the half-line using the Fokas method. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x,t dependence and is given in terms of the spectral functions a(ζ), b(ζ) (obtained from the initial data q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(t)=q_x(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.     

Non-Gaussian effects on quantum entropies

A deduction of generalized quantum entropies within the non-Gaussian frameworks, Tsallis and Kaniadakis, is derived using a generalized combinatorial method and the so-called q and κ calculus. In agreement with previous results, we also show that for the Tsallis formulation the q-quantum entropy is well-defined for values of the nonextensive parameter q lying in the interval [0,2].     

Tsallis’ statistics in the variability of El Niño/Southern Oscillation during the Holocene epoch

Analysis of the Tsallis q-triplet for the variability of El Niño Southern Oscillation (ENSO) index during the Holocene epoch (last 11,000 years) is presented. Three periods are analyzed, 0-7000, 7000-9700, 9700-11,000 years before the present. During the first and the third periods, the q-index values have the expected usual relations between them (q_sens<1<q_stat<q_rel), and in the second one there is an inversion between q_stat and q_rel (q_stat>q_rel).