Two-dimensional Probability Distribution Research Articles

Nucleon exchange in the absence of strong driving forces: The reaction 238U+48Ca at Elab

Mass number, atomic number, energy, and angular distributions have been measured for the projectile-like fragments from the damped reaction $^{238}$U${+}^{48}$Ca at ${E}_{\mathrm{lab}=425}$ MeV. Correlations in the measured two-dimensional probability distribution P(A,Z) are discussed along with the general dissipative features of the damped reaction mechanism. The first and second moments of the observed atomic and mass number distributions are compared with those of other asymmetric reaction systems. Although the $^{238}$U${+}^{48}$Ca system is highly asymmetric, its potential energy surface in the vicinity of the injection point has a nearly zero gradient resulting in a negligible drift in the proton number. The proton drifts observed in reactions of $^{238}\mathrm{U}$ with five different projectiles vary systematically with the potential gradient. The mass and charge data are compared with quantitative predictions by a dynamical reaction model based on nucleon exchange. The theoretical model gives a reasonable account of the mass and charge distributions for the $^{238}$U${+}^{48}$Ca reaction but fails systematically to predict experimental drifts when the driving forces are large.

Effect of a steep gradient in the potential energy surface on nucleon exchange.

Mass number, atomic number, energy, and angular distributions were measured for the projectile-like fragments from the reaction /sup 238/U+/sup 40/Ca at E/sub lab/ = 340 MeV. General features of this reaction are compared to other mass asymmetric systems. Considerable attention is given to the effect of the strong static driving force of the liquid drop (with shell and proximity corrections) potential energy surface on the first and second moments of the observed charge and mass distributions, as well as on associated correlations in the measured two-dimensional probability distribution P(A,Z). Comparison with an appropriate nucleon exchange transport model is made and discrepancies are discussed.

On the nonuniqueness of discretization of two-dimensional probability distribution subject to the maximization of Shannon's entropy (Corresp.)

The maximum entropy method has been successfully applied to a number of scientific and engineering disciplines. A special case of its particular usefulness is the discretization of two-dimensional distribution subject to the maximization of Shannon's entropy. Such a discretization scheme provides a means for deriving a Iow-order approximation of probability distribution for mixed-mode (continuous and discrete) multivariate data. The presence of special cases is shown where the maximum-entropy discretization of two-dimensional probability distribution is not unique.

Two-dimensional probability distribution of attenuation and depolarisation on Earth-space path

Two-dimensional probability distribution of attenuation and crosspolarisation discrimination (XPD) on Earth-space path at 19.5 GHz (circular polarisation) is derived. Moreover, the distribution is approximated by a mathematical function. The distribution is compared with the theoretical relationship between rain-induced attenuation and XPD, which is based on the deformed raindrop model. It is found that the measured attenuation and XPD values are distributed two-dimensionally in an attenuation-XPD plane, and the results cannot be explained by the theoretical relationship. This fact implies that special attention should be paid when the XPD statistics on the Earth-space path are predicted by the theoretical relationship between attenuation and XPD.

Investigations of the two-detector covariance method for the measurement of coupled reactor kinetics parameters

For the measurement of point kinetics parameters of zero power reactors many different methods have been widely used and described in the literature. These include Rossi-α, probability distribution, variance-over-mean, frequency analysis, and polarity correlation of neutron detector signals. Up to now kinetics parameters of a coupled two-point reactor model have been measured only by two noise analysis techniques: Rossi-α (correlation functions) and frequency analysis (power spectral density, coherence functions) using two detectors to obtain space-dependent neutron signals. This paper describes for the first time investigations of two-detector covariance measurements and their application to the determination of coupled kinetics parameters. This technique is compared with the Rossi-α method by theoretical considerations and measurements using identical detector signals in both analysis techniques. Results were also compared to those measured previously at the same reactor by frequency analysis. Theoretical formulas for the Rossi-α experiment and variance and covariance measurements in a symmetrical two-point reactor model were derived. The material properties and neutron lifetimes for each reactor zone were assumed equivalent. The transport time of neutrons from one zone to the other was neglected. The parameters of the model include the decay constant of prompt neutrons and the coupling reactivity of the two core zones. These parameters were determined by least-squares fitting the theoretical curves to the experimental data. The measurements have been performed at the Argonaut Reactor Karlsruhe (ARK) with a symmetrical two-slab core loading for different subcritical levels. Variances and covariances of neutron counts from detectors placed in both core regions were measured in two different ways. In the first method the two-dimensional probability distribution of the number of counts from the detectors was measured using a small digital computer. The computer was used to control the gating times of two specially designed counting registers and to calculate the first and second moments. From that the covariances were calculated in real time and displayed on a CRT as a function of the counting time interval. In the second method the variance and the covariance of the detector pulses were directly measured without using the probability distributions. For this purpose a new analyser based on a special calculational algorithm was developed. This analyser can be built easily with low hardware expense. The results obtained by the Rossi-α and variance method agreed quite well. In contrast to point reactor parameter determination, where Rossi-α and variance techniques are equivalent, the variance and covariance method appear superior to the Rossi-α method in applications to coupled reactors. Kinetics parameters of a two-node reactor model can be determined more accurately by the variance technique. The effect of dead-time losses in the signal channels or in the analyser on the measured variances was also studied. The variance technique was found to be more sensitive to dead-time than the Rossi-α technique. Therefore variance measurements at fast reactors yield incorrect results except special fast electronics is used in the pulse channels.

Two-Dimensional Discrete Probability Distributions

Two-Dimensional Discrete Probability Distributions

Two-Dimensional Probability Structure of Wind-Driven Waves

Measurements of the two-dimensional waveheight probability distribution of wind-driven waves in a model tank suggest that the use of bivariate Gaussian statistics in acoustic scattering computational approximations may be much less realistic than assuming Gaussian behavior in only one dimension.

Statistical properties of velocity derivatives in a turbulent field

High-speed digital computing methods are applied to the study of the statistical behaviour of turbulent velocity derivatives in a nearly isotropic turbulent field downstream of a grid. Higher-order correlations of turbulent velocity gradients, up to the eighth order, are measured. Contrary to the case of velocities, the higher even-order correlations of velocity gradients more clearly evidence the departure from a two-dimensional Gaussian probability distribution. Using non-Gaussian probability distribution laws the relations between different odd-and even-order correlations are obtained and compared with the experimental measurements. The conditions of similarity and isotropy are evaluated for the small-scale structure as evidenced by the behaviour of the turbulent velocity gradients. The concept of intermittency of the small-scale structure is also discussed.

Two-Dimensional Probability Distribution in a Turbulent Field

Two-Dimensional Probability Distribution in a Turbulent Field