Charge order in Magnetite. An LDA+UU study

The electronic structure of the monoclinic structure of Fe$_3$O$_4$ is studied using both the local density approximation (LDA) and the LDA+$U$. The LDA gives only a small charge disproportionation, thus excluding that the structural distortion should be sufficient to give a charge order. The LDA+$U$ results in a charge disproportion along the c-axis in good agreement with the experiment. We also show how the effective $U$ can be calculated within the augmented plane wave methods

Methods of calculation of a friction coefficient: Application to the nanotubes

In this work we develop theoretical and numerical methods of calculation of a dynamic friction coefficient. The theoretical method is based on an adiabatic approximation which allows us to express the dynamic friction coefficient in terms of the time integral of the autocorrelation function of the force between both sliding objects. The motion of the objects and the autocorrelation function can be numerically calculated by molecular-dynamics simulations. We have successfully applied these methods to the evaluation of the dynamic friction coefficient of the relative motion of two concentric carbon nanotubes. The dynamic friction coefficient is shown to increase with the temperature.Comment: 4 pages, 6 figure

Digital adaptive flight controller development

A design study of adaptive control logic suitable for implementation in modern airborne digital flight computers was conducted. Two designs are described for an example aircraft. Each of these designs uses a weighted least squares procedure to identify parameters defining the dynamics of the aircraft. The two designs differ in the way in which control law parameters are determined. One uses the solution of an optimal linear regulator problem to determine these parameters while the other uses a procedure called single stage optimization. Extensive simulation results and analysis leading to the designs are presented

Fusion of neutron rich oxygen isotopes in the crust of accreting neutron stars

Fusion reactions in the crust of an accreting neutron star are an important source of heat, and the depth at which these reactions occur is important for determining the temperature profile of the star. Fusion reactions depend strongly on the nuclear charge $Z$. Nuclei with $Z\le 6$ can fuse at low densities in a liquid ocean. However, nuclei with Z=8 or 10 may not burn until higher densities where the crust is solid and electron capture has made the nuclei neutron rich. We calculate the $S$ factor for fusion reactions of neutron rich nuclei including $^{24}$O + $^{24}$O and $^{28}$Ne + $^{28}$Ne. We use a simple barrier penetration model. The $S$ factor could be further enhanced by dynamical effects involving the neutron rich skin. This possible enhancement in $S$ should be studied in the laboratory with neutron rich radioactive beams. We model the structure of the crust with molecular dynamics simulations. We find that the crust of accreting neutron stars may contain micro-crystals or regions of phase separation. Nevertheless, the screening factors that we determine for the enhancement of the rate of thermonuclear reactions are insensitive to these features. Finally, we calculate the rate of thermonuclear $^{24}$O + $^{24}$O fusion and find that $^{24}$O should burn at densities near $10^{11}$ g/cm$^3$. The energy released from this and similar reactions may be important for the temperature profile of the star.Comment: 7 pages, 4 figs, minor changes, to be published in Phys. Rev.

On the Accuracy of the Semiclassical Trace Formula

The semiclassical trace formula provides the basic construction from which one derives the semiclassical approximation for the spectrum of quantum systems which are chaotic in the classical limit. When the dimensionality of the system increases, the mean level spacing decreases as $\hbar^d$, while the semiclassical approximation is commonly believed to provide an accuracy of order $\hbar^2$, independently of d. If this were true, the semiclassical trace formula would be limited to systems in d <= 2 only. In the present work we set about to define proper measures of the semiclassical spectral accuracy, and to propose theoretical and numerical evidence to the effect that the semiclassical accuracy, measured in units of the mean level spacing, depends only weakly (if at all) on the dimensionality. Detailed and thorough numerical tests were performed for the Sinai billiard in 2 and 3 dimensions, substantiating the theoretical arguments.Comment: LaTeX, 31 pages, 14 figures, final version (minor changes

Decimation and Harmonic Inversion of Periodic Orbit Signals

We present and compare three generically applicable signal processing methods for periodic orbit quantization via harmonic inversion of semiclassical recurrence functions. In a first step of each method, a band-limited decimated periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. In a second step, the frequencies and amplitudes of the decimated signal are determined by either Decimated Linear Predictor, Decimated Pade Approximant, or Decimated Signal Diagonalization. These techniques, which would have been numerically unstable without the windowing, provide numerically more accurate semiclassical spectra than does the filter-diagonalization method.Comment: 22 pages, 3 figures, submitted to J. Phys.

Periodic orbit quantization of the Sinai billiard in the small scatterer limit

We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared to the wave length 2 pi/k. Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant. The limitations of the derived determinant are studied by comparing it to the exact KKR determinant, which we generalize here for the A_1 subspace. With the help of the Ewald resummation method developed for the full KKR determinant we transfer the complex diffractive determinant to a real form. The real zeros of the determinant are the quantum eigenvalues in semiclassical approximation. The essential parameter is the strength of the scatterer c=J_0(kR)/Y_0(kR). Surprisingly, this can take any value between plus and minus infinity within the range of validity of the diffractive approximation kR <<4. We study the statistics exhibited by spectra for fixed values of c. It is Poissonian for |c|=infinity, provided the disk is placed inside a rectangle whose sides obeys some constraints. For c=0 we find a good agreement of the level spacing distribution with GOE, whereas the form factor and two-point correlation function are similar but exhibit larger deviations. By varying the parameter c from 0 to infinity the level statistics interpolates smoothly between these limiting cases.Comment: 17 pages LaTeX, 5 postscript figures, submitted to J. Phys. A: Math. Ge

Dynamical diffraction in sinusoidal potentials: uniform approximations for Mathieu functions

Eigenvalues and eigenfunctions of Mathieu's equation are found in the short wavelength limit using a uniform approximation (method of comparison with a `known' equation having the same classical turning point structure) applied in Fourier space. The uniform approximation used here relies upon the fact that by passing into Fourier space the Mathieu equation can be mapped onto the simpler problem of a double well potential. The resulting eigenfunctions (Bloch waves), which are uniformly valid for all angles, are then used to describe the semiclassical scattering of waves by potentials varying sinusoidally in one direction. In such situations, for instance in the diffraction of atoms by gratings made of light, it is common to make the Raman-Nath approximation which ignores the motion of the atoms inside the grating. When using the eigenfunctions no such approximation is made so that the dynamical diffraction regime (long interaction time) can be explored.Comment: 36 pages, 16 figures. This updated version includes important references to existing work on uniform approximations, such as Olver's method applied to the modified Mathieu equation. It is emphasised that the paper presented here pertains to Fourier space uniform approximation

Semiclassical Theory of Bardeen-Cooper-Schrieffer Pairing-Gap Fluctuations

Superfluidity and superconductivity are genuine many-body manifestations of quantum coherence. For finite-size systems the associated pairing gap fluctuates as a function of size or shape. We provide a parameter free theoretical description of pairing fluctuations in mesoscopic systems characterized by order/chaos dynamics. The theory accurately describes experimental observations of nuclear superfluidity (regular system), predicts universal fluctuations of superconductivity in small chaotic metallic grains, and provides a global analysis in ultracold Fermi gases.Comment: 4 pages, 2 figure

Nonperiodic Orbit Sums in Weyl's Expansion for Billiards

Weyl's expansion for the asymptotic mode density of billiards consists of the area, length, curvature and corner terms. The area term has been associated with the so-called zero-length orbits. Here closed nonperiodic paths corresponding to the length and corner terms are constructed.Comment: 8 pages, 2 figure