Sixth-order Coefficients Research Articles

Derivation, characterization, and application of complete orthonormal sequences for representing general three-dimensional states of residual stress

Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to their complex origins, it is useful to develop functions that can be linearly combined to represent any sufficiently regular residual stress field. In this work, we develop orthonormal sequences that span the set of all square-integrable residual stress fields on a given three-dimensional region. These sequences are obtained by extremizing the most general quadratic, positive-definite functional of the stress gradient on the set of all sufficiently regular residual stress fields subject to a prescribed normalization condition; each such functional yields a sequence. For the special case where the sixth-order coefficient tensor in the functional is homogeneous and isotropic and the fourth-order coefficient tensor in the normalization condition is proportional to the identity tensor, we obtain a three-parameter subfamily of sequences. Upon a suitable parameter normalization, we find that the viable parameter space corresponds to a semi-infinite strip. For a further specialized spherically symmetric case, we obtain analytical expressions for the sequences and the associated Lagrange multipliers. Remarkably, these sequences change little across the entire parameter strip. To illustrate the applicability of our theoretical findings, we employ three such spherically symmetric sequences to accurately approximate two standard residual stress fields. Our work opens avenues for future exploration into the implications of different sequences, achieved by altering both the spatial distribution and the material symmetry class of the coefficient tensors, toward specific objectives.

Higher-order symmetry energy and neutron star core-crust transition with Gogny forces

We study the symmetry energy and the core-crust transition in neutron stars using the finite-range Gogny nuclear interaction and examine the deduced crustal thickness and crustal moment of inertia. We start by analyzing the second-, fourth- and sixth-order coefficients of the Taylor expansion of the energy per particle in powers of the isospin asymmetry for Gogny forces. These coefficients provide information about the departure of the symmetry energy from the widely used parabolic law. The neutron star core-crust transition is evaluated by looking at the onset of thermodynamical instability of the liquid core. The calculation is performed with the exact (i.e., without Taylor expansion) Gogny EoS for the core, and also with its Taylor expansion in order to assess the influence of isospin expansions on locating the inner edge of neutron star crusts. It is found that the properties of the core-crust transition derived from the exact EoS differ from the predictions of the Taylor expansion even when the expansion is carried through sixth order in the isospin asymmetry. Gogny forces, using the exact EoS, predict the ranges $0.094 \text{ fm}^{-3} \lesssim \rho_t \lesssim 0.118\text{ fm}^{-3}$ for the transition density and $0.339 \text{ MeV fm}^{-3} \lesssim P_t \lesssim 0.665 \text{ MeV fm}^{-3}$ for the transition pressure. The transition densities show an anticorrelation with the slope parameter $L$ of the symmetry energy. The transition pressures are not found to correlate with $L$. Neutron stars obtained with Gogny forces have maximum masses below $1.74M_\odot$ and relatively small moments of inertia. The crustal mass and moment of inertia are evaluated and comparisons are made with the constraints from observed glitches in pulsars.

The temperature, concentration phase diagram for Tb1−xDyxZn: Failure of crystal field theory

We have calculated the free energy of a crystal field model designed to reproduce the magnetic temperature, composition phase diagram for Tb1−xDyxZn alloys. Using fourth- and sixth-order coefficients consistent with single-ion theory, it was found to be impossible to reproduce the experimental phase boundaries. A reasonable fit to the entire diagram was obtained when the coefficients were chosen to be of the same sign for Dy and Tb, rather than of opposite sign, as required by single-ion theory. To explain this discrepancy, a model is introduced which includes a contribution to the crystal field coming from quadrupole–quadrupole interactions. It is argued that the reason for the success of the calculation using coefficients of the same sign is directly related to the dependence of the strength of the quadrupole coupling on the square of the second-order Stevens factor.

Geometrical analysis of a lens-like medium with higher-order aberration

The propagation of light rays in a slab-waveguide whose dielectric constant varies radially with a quadratic law but has fourth and sixth order terms in it is described. Exact solutions consisting of Jacobi elliptic functions are shown. The exact ray trajectory and delay time of an optical pulse, therefore, can be obtained. The results show that the optimum fourth-order and sixth-order coefficients are near 2/3 and 17/45, respectively.

Short-distance behavior of quantum electrodynamics and the Callan-Symanzik equation for the photon propagator

The short-distance behavior of the photon propagator is discussed within the context of the corresponding Callan-Symanzik equation. The Callan-Symanzik function β(α) is calculated in perturbation theory up to sixth order. We find β(α) = 2 3 α π + 1 2 α π 2 − 121 144 α π 3 + O α π 4 . The simplicity of this result is to be contrasted with a corresponding perturbation theory calculation of the Gell-Mann-Low function ψ( z), whose sixth-order coefficient contains the transcendental ζ(3) (the Riemann zeta function of argument three). A mechanism of cancellations in the calculation of β(α) has been found, and we prove its validity to all orders in perturbation theory.

The effect of d electrons on crystal field potentials in rare earth metals and dilute alloys

The effect of 5d character on the crystalline electric field in noble metal rare earth alloys and in pure rare earths has been investigated. The change in sign of the fourth order CEF coefficient in the noble metal alloys can be interpreted in terms of this d character. From symmetry this d like component has no effect on the sixth order coefficients whose varying magnitude in these alloys can be explained using the f like contribution. In the pure rare earth metals it is likely that d character will predominate.

The Current Status of the LeptongFactors

The past three years have seen new measurements of the $g$-factor anomalies of the free leptons (${e}^{\ifmmode\pm\else\textpm\fi{}}, {\ensuremath{\mu}}^{\ifmmode\pm\else\textpm\fi{}}$). At the same time there have been several major theoretical advances including a quantum electrodynamic calculation of the sixth-order coefficient of the electron anomaly. In this article we review these recent developments in detail, however we also: (1) present a historical overview of $g$-factor measurements and their interpretation since the early days of quantum theory, (2) discuss several experiments which are currently being attempted but which have not yet yielded results of high precision, (3) attempt to predict the most likely course of $g$-factor research in the near future. An extensive Bibliography and Tables summarizing both historical events and current work are included.