Invariant Expansion Research Articles

Rapidity dependence of thermal dileptons resulting from hadronizing quark-gluon matter with finite baryon charge

The influence of a non-vanishing baryon charge on the rapidity distribution of dileptons produced in ultrarelativistic heavy-ion collisions is studied. We employ a frozen motion model with scaling invariant expansion of the hadronizing quark-gluon plasma as well as a realistic rapidity distribution of secondary particles (i.e., pions and baryons) expected for RHIC energies. We demonstrate a sizeable suppression of the thermal dilepton yield at large rapidities due to the finite baryon density. This affects the most favorable rapidity window for using dileptons for the diagnostic of the quark-gluon plasma and the early temperature distribution of the hot reaction zone. To discriminate the thermal dileptons from Drell-Yan background we propose to utilize the dilepton yield scaled suitably by the pion multiplicity as function of rapidity.

A phase-space technique for the perturbation expansion of Schrödinger propagators

A perturbation theory for Schrödinger and heat equations that is based on phase-space variables is developed. The Dyson series representing the evolution kernel is described in terms of two basic classical quantities: the free classical motion along flat space geodesics and the Green function for the Jacobi operator in phase space. Further, for problems with Abelian interactions it is demonstrated that the perturbation theory may be summed to all orders yielding an exponentiated connected graph description for the evolution kernel. Connected graph representations provide an efficient method of constructing various semiclassical approximations wherein expansion coefficients are directly determined by explicit cluster integrals. This type of application is discussed for the case of Schrödinger and heat equations with external electromagnetic fields. Detailed expressions for coefficients are obtained for both the gauge invariant large mass expansion as well as the short time Schwinger–DeWitt expansion. Finally it is shown how to apply this phase-space method so that it incorporates a recently proposed covariant perturbation theory.

Longevity of the quark-gluon plasma and the mixed phase from intensity interferometry of high energy photons

Two-photon intensity interferometry is shown to provide an accurate measurement of the lifetime of the quark-gluon plasma created in ultra-relativistic heavy ion collisions via the difference of outward and sideward correlation radii. Under the assumption of a longitudinal, boost invariant expansion of the plasma, we obtain analytical expressions for the correlations from the quark-gluon plasma phase. A 3 + 1 dimensional expansion of the plasma along with a first order phase transition to hadrons is next considered, and leads to a source with two characteristic lifetimes, one of the quark-gluon plasma phase, and the other for the longer lived mixed phase. This may even help us to experimentally determine the order of the phase transition.

Viscohyperelastic Modeling of Rubber Vulcanizates

Abstract The use of internal hyperelastic solids for modeling viscoelastic deformations of rubber vulcanizates is reviewed. The model is applied in one dimension to viscoelastic uniaxial tension and uniaxial shear experiments. Step-strain relaxation tests are used to determine the model's parameters. A hyperelastic energy function, which represents the sum of the internal solids' energy functions, is obtained by least squares fitting a constrained third-order invariant expansion of the Rivlin function to the difference between the step-strain stresses and the relaxed stresses (the standard hyperelastic solid's stresses). The difference energy function is split into two parts and relaxation parameters (related to the rate of change of the internal solids' reference lengths) are selected so that numerically simulated step-strain relaxation stresses approximate the experimental values (at approximately 50 ms). The model is then used to predict the experimental results from a different type of test, cyclic strain data, at three different strain rates (cyclic frequencies). Increased stress due to increased strain rate was indicated by the model for large strains.

Numerical solution of the RHNC theory for fluids of non-spherical particles near a uniform planar wall

An efficient algorithm, a hybrid of the Picard-type and Newton-Raphson (NR) methods, is developed for solving the reference hypernetted-chain (RHNC) theory for non-spherical particles near a uniform planar wall. The basic idea of the algorithm is also applicable to non-spherical particles near a large macroparticle at infinite dilution. The problem of dipolar hard spheres near a hard wall is chosen here as an example system. A feature of the algorithm is that the Jacobian matrix is determined in the bulk calculation and serves as part of the input data for the wall calculation. Hence, the convergence is achieved in only a single iteration in the inner NR loop regardless of the initial values of the iteration variables. It is shown that the great advantage of the constant Jacobian matrix can be preserved even for general cases of other particle-particle and wall-particle interactions. In the course of the study, we encounter a singular behaviour of the RHNC theory for bulk dipolar hard spheres. Keeping the reduced density, we continue to increase the reduced dipole moment. Then, at a certain point the projections of the rotational invariant expansion of the total correlation function h mm0(r) (m ≠ 0; m = 1,2,3, …) become infinitely long ranged, which causes divergence of their Fourier transforms at zero wave-number, ħ mm0(0). The static dielectric constant and the heat capacity also diverge at that point. The singular behaviour implies the growth of the strong, long-ranged orientational order favouring parallel configuration, and represents a stability limit of the isotropic phase.

Additive anisotropic interactions in molecular liquids and liquid crystals

Formulae for a SO(3)-invariant expansion of the pair-molecular interaction-energy are derived. Two different cases, of exponential potential and of inverse-power potential, are considered for molecules of arbitrary shape. Methods for the determination of the generalized multipole components for molecules are developed on the base of transformation formulae for rotation and translation of a coordinate system. The importance of accounting the difference between the general expansion and that of a conventional multipole type is illustrated on the base of simulation data for mesogen substance PCH-5.

Conduction subbands in a GaAs/AlxGa1-xAs quantum well: Comparing different k

The energy dispersion of the conduction subbands in a GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As superlattice is calculated by using a k\ensuremath{\cdot}p Hamiltonian that includes different number of bands. The most accurate model includes the ${\mathrm{\ensuremath{\Gamma}}}_{6}^{\mathit{c}}$, ${\mathrm{\ensuremath{\Gamma}}}_{7}^{\mathit{v}}$, ${\mathrm{\ensuremath{\Gamma}}}_{8}^{\mathit{v}}$, ${\mathrm{\ensuremath{\Gamma}}}_{7}^{\mathit{c}}$, and ${\mathrm{\ensuremath{\Gamma}}}_{8}^{\mathit{c}}$ bands. The resulting subband dispersion is compared with that obtained when the coupling of the ${\mathrm{\ensuremath{\Gamma}}}_{6}^{\mathit{c}}$ band with the ${\mathrm{\ensuremath{\Gamma}}}_{7}^{\mathit{v}}$ and ${\mathrm{\ensuremath{\Gamma}}}_{8}^{\mathit{v}}$ and/or ${\mathrm{\ensuremath{\Gamma}}}_{7}^{\mathit{c}}$ and ${\mathrm{\ensuremath{\Gamma}}}_{8}^{\mathit{c}}$ bands is neglected. We also consider the 2\ifmmode\times\else\texttimes\fi{}2 k\ensuremath{\cdot}p Hamiltonian with terms up to the order ${\mathit{k}}^{4}$. The subband dispersions are analyzed quantitatively by fitting the numerical result to the analytical expression obtained with the invariant expansion technique. The subbands are found to be significantly different with the various models. The differences are ascribed to the different bulk band dispersions obtained with different k\ensuremath{\cdot}p Hamiltonians and to the variation of the band parameters in the well and barrier materials.

Exponential cluster solutions to quantum transport equations

Connected graph methods are used to solve the transport equation arising in a semiclassical representation of the Schrodinger propagator K(t,s,x,y). For a spinless quantum system whose Hamiltonian is a smooth position- and momentum-dependent perturbation of the Laplacian in Rd, exponential cluster expansions for K are obtained. These formally exact solutions to the time-dependent Schrodinger equation employ explicit graphically-determined derivatives of the Hamiltonian's symbol, integrated along geodesics. For the special case of particles interacting with external electromagnetic fields, the propagator's gauge invariant derivative expansion coefficients are determined in closed form in terms of the Lorentz force. A tree graph formula for Hamilton's principal function is extracted from this result.

Magnetic structure in FeRh from constrained total-energy calculations

The volume dependence of antiferromagnetic and ferromagnetic total-energy solutions derived from constrained total-energy band calculations is shown to explain the pressure and magnetic field effects on the antiferromagnetic to ferromagnetic transition in FeRh. We show that zero-point energy corrections are necessary to obtain reasonable agreement with experimental critical pressures and magnetic fields, and that the difference in electronic specific heats of the different magnetic states is small and does not play a role in the transition. Lattice constants of the magnetic states are inverted with respect to Invar, leading to enhanced rather than invariant thermal expansion near the transition, so that FeRh can be classified as an “anti-Invar” system.

Multi-scale semiclassical approximations for Schrödinger propagators on manifolds

Exponential representations of the Schrödinger propagator are constructed for an interacting quantum system on a semi-Riemannian manifold M. For a local fundamental solution of the time dependent Schrödinger equation, the higher-order WKB approximation is re-expanded in the scaling parameter ε responsible for the gauge invariant derivative expansion. The geometrical methods which establish the consistency between the WKB and ε expansions involve a perturbative analysis of the classical dynamics on M which employs the Green function of the inhomogeneous geodesic deviation equation. The analysis applies when both propagator arguments lie in a region which is convex with respect to the classical motion. A multi-scale analysis in the parameters ħ, ε, charge λ and time displacement Δ t is performed, leading to simple basic coefficient functions and their recurrence relations. These propagator representations display explicitly the interplay between the local geometry of the manifold and the effects of the background electromagnetic fields.

TRS: a program to calculate Landau levels and direct dipole transitions in uniaxially stressed semiconductors

Quantum resonances in the valence bands of semiconductors under uniaxial stress provide very detailed information on the band parameters if the experimental data can be analyzed on the basis of an adequate theoretical model. Trebin and Rössler developed such a model for narrow-gap semiconductors with a zincblende lattice and applied it to InSb. Using their theoretical results they wrote a program for the numerical evaluation of energy eigenvalues, wave functions and oscillator strengths for direct inter- and intraband dipole transitions. It is based on an effective Hamiltonian constructed by invariant expansion and describes the Landau levels in an eightfold space of valence and conduction bands when uniaxial stress and magnetic field are applied along the [001] direction. All eigenstates are evaluated for any three values of wave vector, magnetic field and stress. Recently, Schmitz made several improvements and additions to this program including a complete user interface making the program easy to use, adaptable to a wide variety of computers, and allowing calculations done systematically.

Symmetry and strain-induced effects at the W point of the Brillouin zone of face-centered-cubic crystals.

The invariant expansions of the effective-mass Hamiltonian at the W point of fcc crystals including strain interactions are presented for the space groups Fm3m (${\mathit{O}}_{\mathit{h}}^{5}$), Fd3m (${\mathit{O}}_{\mathit{h}}^{7}$), and F4\ifmmode\bar\else\textasciimacron\fi{}3m (${\mathit{T}}_{\mathit{d}}^{2}$). The intervalley and intravalley splittings induced by uniaxial stress along high-symmetry directions are derived for both phonon and electron spectra. Relevant parameters for the electronic bands of germanium near W are calculated with the ab initio linear-muffin-tin-orbital method.

An asymptotic analysis of quantum evolution with electromagnetic fields

A singular perturbation expansion of solutions to the Schrödinger initial value problem is constructed using an approximate propagator. For a nonrelativistic quantum system interacting with time-dependent external electromagnetic fields, this approximate propagator defines a gauge invariant semiclassical expansion that is realized by large mass scaling. The asymptotic nature of this approximation is established by constructing error estimates that bound the Hilbert space norm difference between the exact and approximate evolved states. The maximum order of the approximation is determined explicitly as a function of the number of derivatives supported by the scalar and vector potentials. The asymptotic expansion is obtained when the configuration space Ω=Rd, and also for problems where Ω is a proper subset of Rd and the self-adjoint Hamiltonian is defined using a supplementary boundary condition—typically Dirichlet or periodic.

Time-dependentX- andY-functions for an isotropically scattering neutron transport problem in a finite slab

Expressions for time-dependentX- andY-functions for a one-speed neutron transport problem in a finite slab have been derived using a technique combining invariant imbedding method and eigenfunction expansion method. The atmosphere has been considered to scatter isotropically.

Gauge invariant asymptotic expansion of schrödinger propagators on manifolds

A semiclassical technique is developed for approximating the time evolution of quantum systems on semi-Riemannian manifolds in the presence of external electromagnetic fields. In particular, a geometrically covariant and gauge covariant formal asymptotic expansion of the Schrödinger propagator is derived. The theory is applicable not only to spinless nonrelativistic quantum systems with constraints, but also to general-relativistic scalar field theory through the Schwinger-DeWitt equation. The basic classical objects in this approach are the geodesics of the manifold. The expansion—in the derivatives of the gravitational and electromagnetic fields—is valid when both propagator arguments lie in a geodesically convex region. The zeroth order approximation consists of the wkb propagator for free quantum evolution on the manifold, modified by a gauge-integral phase factor which carries all the gauge structure of the exact propagator. Higher order expansion coefficients are gauge invariant solutions of a one-dimensional integral recurrence relation derived from geodesic transport. By using a Jacobi field and a Green function associated with the geodesic deviation equation, the expansion coefficients can be put into an explicit form. The first one is computed in detail and manifestly displays the effects of gravity and electromagnetism, both separately and in combination. The expansion derived here generalizes the well-known short time expansion in several ways.

Fine structure of quantum well excitons

The enhancement of the exciton binding energy due to confinement in a quantum well is accompanied by an even stronger enhancement of the fine structure splitting caused by the electron-hole exchange interaction. We present a systematic grouptheoretical formulation of the exchange Hamiltonian by means of an invariant expansion and perform quantitative calculations of the exchange splitting which take into account the heavy-hole light-hole mixing effects in the hole and exciton states.

Reduction of invariant differential operators and expansions of conical distributions for $SO_0(n, 1)$

Reduction of invariant differential operators and expansions of conical distributions for $SO_0(n, 1)$

Invariant expansions of the radial distribution function for fluids of hard convex bodies

The radial distribution functions (RDF) for systems consisting of a single nonspherical convex body in a fluid of spheres and for a pure fluid of nonspherical convex bodies are expressed in terms of the minimum surface-to-surface separation and a set of angles derived from the surface normal. The orthogonal function expansion of the anisotropic RDF, g, has the property that the first anisotropic contribution vanishes at zero density and is directly proportional to the nonsphericity of the particle; however, the resulting convex body orthogonal polynomials depend on the surface-to-surface separation. An invariant expansion of the product of the Jacobian and the RDF, S12g, has anisotropic expansion coefficients even at zero density, but the expansion functions are the spherical harmonics (independent of the surface-to-surface coordinate). Monte Carlo simulations are conducted on the two dimensional system consisting of a single convex ellipse in a bath of circular disks. A comparison is made between the center-to-center expansion of the RDF, an expansion based on the orthogonal function expansion, and the S12g (or Kabadi–Steele) expansion.

Quantum systems with external electromagnetic fields: The large mass asymptotics

The large mass asymptotics of the quantum evolution problem for a system of charged particles that mutually interact through scalar fields and couple to an arbitrary time-varying external electromagnetic field is rigorously described. If K(x,t; y,s;m) denotes the coordinate space propagator (time evolution kernel) of this system, the singular perturbation behavior of K as mass m→∞ is expressed in terms of a gauge invariant asymptotic expansion. In terms of the external fields and interparticle interactions, this expansion provides a nonperturbative approximation for the propagator K that is valid for all particle coordinates x, y and for finite time displacements t−s. For the class of analytic scalar and vector fields that are defined as Fourier transforms of time-dependent measures, the existence of this asymptotic series for K in powers of (m)−1 is established for both real and complex masses. Explicit bounds for the error term are obtained and a manifestly gauge invariant transport recurrence relation is derived that uniquely determines all the coefficient functions of the asymptotic series. The small time asymptotic expansion of K is shown to be embedded within the large mass expansion.

Equation des chaînes hypertressées adaptée aux potentiels anisotropes répulsifs à courte portée. Résultats préliminaires pour un fluide de bâtonnets

We study the fluids of non spherical rigid molecules which are generated by site distributions. The interactions between two sites located on different molecules are assumed to be soft and isotropic. We present a general and efficient numerical method to compute the coefficients of the rotational invariant expansion of the anisotropic intermolecular potentials obtained as the superpositions of the site-site interactions. A new closure HNCAR of the integral equation theory is proposed. This closure, which is of the HNC type at long distances, A-mplifies the R-epulsive effects of the soft shape intermolecular potentials which become able to efficiently prevent the overlap of the molecules, even when strong attractive forces are present. The formalism is applied to fluids of rods made of repulsive sites. The structure and the equation of state of these fluids are studied using the HNCAR theory and the results obtained are found to improve upon those given by the HNC and SMSA approximations Etude des fluides de molecules rigides non spheriques, engendres par des distributions de sites, en supposant que deux sites situes sur des molecules differentes sont couples par des interactions isotropes nulles. Presentation d'une methode numerique pour calculer les coefficients du developpement en serie d'invariants rotationnels des potentiels intermoleculaires anisotropes resultant de la superposition de ces interactions site-site. Developpement d'une equation de type HNC adaptee a des potentiels intermoleculaires contenant une partie repulsive a courte portee pour l'application a un fluide de bâtonnets