Eigenvalue Separation Research Articles

On relationships among effective potential functions: 2,5-dihydrofuran, s-tetrazine, 1,4-dioxadiene, cyclopentene, and carbon suboxide

Using Van Vleck perturbation theory, a relationship between two-dimensional potential functions in ring-puckering and ring-twisting coordinates and effective one-dimensional ring-puckering coordinates is demonstrated. It is shown that effective one-dimensional potential functions can be derived for 2,5-dihydrofuran and s-tetrazine which have essentially the same eigenvalue separations as the two-dimensional potential functions previously derived. More useful is the fact that this procedure may be reversed and two-dimensional potential surfaces have been derived for 1,4-dioxadiene and cyclopentene. In mass weighted coordinates the potential function for dioxadiene is V (Q1,Q2) =28.49 Q14+24.00 Q12 +2555.2 Q22+294.6 Q12Q22, where V is in cm−1, Q in U1/2 Å. Q1 is a mass weighted ring-puckering coordinate and Q2 is a mass weighted ring-twisting coordinate. The potential function for cyclopentene is V (Q1,Q2) =51.50 Q14−217.4 Q12+2262.4 Q22 −9.10 Q12Q22 yielding a barrier to planarity of 229 cm−1 essentially unchanged from the barrier of 232 cm−1 reported earlier from the one-dimensional potential function. This same formalism has been used to correlate the potential functions for the large amplitude bending mode, ν7, of carbon suboxide in the ground state and several excited states of other modes.

Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories

Quantum stochasticity (the nature of wave functions and eigenvalues when the short-wave-limit Hamiltonian has stochastic trajectories) is studied for the two-dimensional Helmsholtz equation with stadium boundary. The eigenvalue separations have a Wigner distribution (characteristic of a random Hamiltonian), in contrast to the clustering found for a separable equation. The eigenfunctions exhibit a random pattern for the nodal curves, with isotropic distribution of local wave vectors.

Steady state of stiff linear kinetic systems by a Markov matrix method

The absence of detailed balance in systems containing pseudo-first-order reactions can cause the evaluation of steady-state concentrations to be a difficult computational problem. if the rate constants differ by many orders of magnitude, direct solution for these concentrations from the matrix of rate constants is not practical. The embedded Markov chain generated by the state-changes of one molecule has a transition matrix whose better separation of eigenvalues makes it more suitable for analysis than the matrix of rate constants. Solution for an eigenvector of this matrix by the method of inverse iteration provides an efficient means for obtaining steady-state concentrations.

Boundary Value Control of the Wave Equation in a Spherical Region

In this paper we study controllability problems for the wave equation \[\frac{{\partial ^2 w}}{{\partial t^2 }} - \sum_{j = 1}^n {\frac{{\partial ^2 w}}{{\left( {\partial x^j } \right)^2 }}} ,\qquad t \geqq 0,\quad x \in \Omega ,\] where $\Omega $ is a spherical region in $R^n $. The control force f enters in the boundary condition \[\frac{{\partial w}}{{\partial v}} = f,\] assumed to hold for $t \geqq 0$, $x \in \Gamma = \partial \Omega $. Our main result is that all “finite energy” initial states can be steered to the zero state in time $\tau $, using a control $f \in L^2 (\Gamma \otimes [0,\tau ])$, provided $\tau > 2$.Beginning with standard existence results for solutions of the wave equation, we show the control problem to be equivalent to a collection of trigonometric moment problems. These are solved using the theory of nonharmonic Fourier series together with certain results concerning the separation of eigenvalues of the Helmholtz operator in $L^2 (\Omega )$ with the Neumann boundary condition. ...

Interpretation of Coherence Function Measurements in Zero-Power Coupled-Core Reactors

New techniques for inferring several reactor kinetic parameters from frequency-domain noise analysis measurements in coupled-core reactors are developed and utilized. These parameters are the eigenvalue separation, delayed-neutron fraction, and, independently, the prompt-neutron lifetime. Reactor characteristics and detector placement criteria necessary for obtaining the parameters are discussed. Comparison with other time-domain noise techniques is undertaken and the merits and shortcomings of each are reviewed.

Separation of Eigenvalues of the Wave Equation for the Unit Ball in RN

It is shown that π is the infinium gap between the consecutive square roots of the eigenvalues of the wave equation in a hypespherical domain for both the Neumann (free) and the full range of mixed (elastic) homogeneous boundary conditions. Previous literature contains the same information apparently only for the Dirichlet (fixed) boundary condition. These square roots of the eigenvalues are the zeros of solutions of a differential equation in Bessel functions (first kind) and their first derivatives. The infinium gap is uniform for Bessel functions of orders x ≥ ½ (as well as for x = 0). The intervals between the roots are actually monotone decreasing in length. These results are obtained by interlacing zeros of Bessel and associated functions and comparing their relative displacements with oscillation theory. If Wl denotes the lth positive root for some fixed order x, the minimum gap property assures that {exp(±iwlt|l = 1, 2,...} form a Riesz basis in L2(0, τ) for τ > 2. This has application to the problem of controlling solutions of the wave equation by controlling the boundary values.

Noise and Transient Kinetics Experiments and Calculations for Loosely Coupled Cores

A series of space-time reactor kinetics experiments has been performed in the Solid Homogeneous Assembly. The ground rules were the following: The cores were to be easy to model statically using few-group diffusion theory.The dynamic behavior was to be one dimensional.Significant flux tilt and delayed neutron flux tilt holdback effects were to be present.The perturbation was to be essentially a negative step change in reactivity.The flux distributions, absorber worths, and eigenvalue separation of each core were to be measurable in order to use this information to verify the core modeling process.The adequacy of few-group diffusion theory was checked using both one-dimensional multigroup (fast and thermal) and two-dimensional few-group transport theory calculations. The few-group diffusion theory model was solved in a cylindrical approximation using a finite difference code, and in correct 3D geometry using a synthesis code; the latter did an excellent job of predicting criticality and matching foil activation traverses made using the 55Mn (n, γ) thermal and 115In (n, n’) threshold reactions. The eigenvalue separation of each core was measured using both a static flux tilt method and a two-detector noise correlation method; the results agreed well with calculated values, indicating that these new experimental methods can provide an accurate determination of the eigenvalue separation of a core.The transient response of one of the cores to a step perturbation was compared to the response calculated using the space-time synthesis model. In general, it has been found that the transient response of the core can be predicted accurately provided that the core is well modeled statically, including the perturbation worth and the eigenvalue separation, and provided that the effective delayed neutron fraction is properly evaluated. This tends to verify the adequacy of existing methods for computing spatially dependent transients in cores describable by few-group diffusion theory.