Continuum (set theory)

About: Continuum (set theory) is a research topic. Over the lifetime, 3838 publications have been published within this topic receiving 97654 citations.

Effects of Configuration Interaction on Intensities and Phase Shifts

Abstract: The interference of a discrete autoionized state with a continuum gives rise to characteristically asymmetric peaks in excitation spectra. The earlier qualitative interpretation of this phenomenon is extended and revised. A theoretical formula is fitted to the shape of the $2s2p^{1}P$ resonance of He observed in the inelastic scattering of electrons. The fitting determines the parameters of the $2s2p^{1}P$ resonance as follows: $E=60.1$ ev, $\ensuremath{\Gamma}\ensuremath{\sim}0.04$ ev, $f\ensuremath{\sim}2 \mathrm{to} 4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. The theory is extended to the interaction of one discrete level with two or more continua and of a set of discrete levels with one continuum. The theory can also give the position and intensity shifts produced in a Rydberg series of discrete levels by interaction with a level of another configuration. The connection with the nuclear theory of resonance scattering is indicated.

The stress-dilatancy relation for static equilibrium of an assembly of particles in contact

Abstract: The dilatancy and strength of an assembly of individual particles in contact when subjected to a deviatoric stress system is found to depend on the angle of friction $\phi\_\mu$ between the particle surfaces, on the geometrical angle of packing, $\alpha$, and on the degree of energy loss during remoulding. The Mohr-Coulomb criterion of failure which is strictly applicable to a continuum is shown not to have general application to a discontinuous assembly of particles. A theoretical and experimental study of ideal assemblies of rods and uniform spheres establishes expressions for the relation between the rate of dilatancy and the maximum stress ratio for any ideal packing. The solution is extended to the case of a random assembly of irregular particles by investigating the conditions under which the mass dilates such that the rate of internal work absorbed in frictional heat is a minimum. Experiments on random masses of steel, glass, and quartz in which all the physical properties are measured independently show that the minimum energy criterion is closely obeyed by highly dilatant dense over-consolidated and reloaded assemblies throughout deformation to failure. An additional rate of energy has to be supplied to account for losses due to rearranging of loose packings, when the value of $\phi$ to satisfy the theory increases to $\phi\_f$ by an amount dependent on the degree of remoulding. The external stresses applied to an assembly are to be integrated over the $\alpha$-plane defined with reference to figure 14(a) as a plane of repetition of pattern over which the particles interlock, and the resulting forces are to be in equilibrium for sliding on particle interfaces at (45-$\frac{1}{2}\phi_f$) to the direction of the major principal stress. For the special case of no volume change these two planes are identical and the solution agrees then with that based on the Mohr-Coulomb theory. The well-known slip plane in drained discontinuous assemblies is proved to be the result of failure and nothing whatsoever to do with the peak strength. The findings are discussed in the light of previous contributions to the subject.

Intensity of Optical Absorption by Excitons

Abstract: The intensity of optical absorption close to the edge in semiconductors is examined using band theory together with the effective-mass approximation for the excitons. Direct transitions which occur when the band extrema on either side of the forbidden gap are at the same K, give a line spectrum and a continuous absorption of characteristically different form and intensity, according as transitions between band states at the extrema are allowed or forbidden. If the extrema are at different K values, indirect transitions involving phonons occur, giving absorption proportional to ${(\ensuremath{\Delta}E)}^{\frac{1}{2}}$ for each exciton band, and to ${(\ensuremath{\Delta}E)}^{2}$ for the continuum. The experimental results on ${\mathrm{Cu}}_{2}$O and Ge are in good qualitative agreement with direct forbidden and indirect transitions, respectively.

Quantum-size effects of interacting electrons and holes in semiconductor microcrystals with spherical shape

Abstract: Quantum-size effects of an electron-hole system confined in microcrystals of semiconductors are studied theoretically with the spherical-dielectric continuum model. An extensive numerical calculation for the eigenvalue problem is carried out by Ritz's variational technique. The motional state of the lowest level is classified into three regimes: the regime of exciton confinement for R/${a}_{B}^{\mathrm{*}}$\ensuremath{\gtrsim}4, the regime of individual particle confinement for R/${a}_{B}^{\mathrm{*}}$\ensuremath{\lesssim}2, and the intermediate regime for 2\ensuremath{\lesssim}R/${a}_{B}^{\mathrm{*}}$\ensuremath{\lesssim}4, where R is the radius of the quantum well and ${a}_{B}^{\mathrm{*}}$ is the exciton Bohr radius. In the region R/${a}_{B}^{\mathrm{*}}$\ensuremath{\gtrsim}4, the high-energy shift of the lowest exciton state is described by the rigid-sphere model of the exciton quite well, which takes into account the spatial extension of the relative motion of the electron and the hole. The oscillator strength of the interband optical transition changes dramatically across the region 2\ensuremath{\lesssim}R/${a}_{B}^{\mathrm{*}}$\ensuremath{\lesssim}4. The metamorphosis of the absorption spectrum is shown as a function of R/${a}_{B}^{\mathrm{*}}$ and compared with the experimental data.

The Continuum Random Tree III

Abstract: Let $(\mathscr{R}(k), k \geq 1)$ be random trees with $k$ leaves, satisfying a consistency condition: Removing a random leaf from $\mathscr{R}(k)$ gives $\mathscr{R}(k - 1)$. Then under an extra condition, this family determines a random continuum tree $\mathscr{L}$, which it is convenient to represent as a random subset of $l_1$. This leads to an abstract notion of convergence in distribution, as $n \rightarrow \infty$, of (rescaled) random trees $\mathscr{J}_n$ on $n$ vertices to a limit continuum random tree $\mathscr{L}$. The notion is based upon the assumption that, for fixed $k$, the subtrees of $\mathscr{J}_n$ determined by $k$ randomly chosen vertices converge to $\mathscr{R}(k)$. As our main example, under mild conditions on the offspring distribution, the family tree of a Galton-Watson branching process, conditioned on total population size equal to $n$, can be rescaled to converge to a limit continuum random tree which can be constructed from Brownian excursion.