ALPHA DECAY OF SPHEROIDAL NUCLEI

1958 ◽  
Vol 36 (7) ◽  
pp. 944-962 ◽  
Author(s):  
E. M. Pennington ◽  
M. A. Preston
Keyword(s):  
Distribution Function ◽  
Probability Density ◽  
Symmetry Axis ◽  
Electronic Computer ◽  
Alpha Decay ◽  
Alpha Particles ◽  
Nuclear Surface ◽  
Barrier Penetration ◽  
Spherical Nuclei ◽  
Spheroidal Coordinates

A system of coupled differential equations relates the amplitudes of alpha particle waves emitted from a spheroidal nucleus described by the Bohr–Mottelson model. These equations in spheroidal coordinates have been solved for five even–even nuclides with the aid of the electronic computer FERUT. There are four possible cases for each nuclide which are consistent with the boundary conditions. The solutions of the equations are used to calculate the probability density of alpha particles on the nuclear surface for each case. For Case I the peak in the distribution function shifts from the nuclear symmetry axis to the equator with increasing mass number. The probability density for Case II is always peaked between the symmetry axis and the equator, while for Cases III and IV it is always peaked strongly at the equator. The change in the distribution function with increasing distance from the nucleus is considered for a typical case. Barrier penetration factors are calculated and found to differ from those for spherical nuclei by factors of the order of 2 or 3. Comparison with the calculations of an approximation method of Fröman is made for one nuclide.

A systematic calculation of alpha decay half-lives using a new approach for barrier penetration probability

2016 ◽  
Vol 25 (09) ◽  
pp. 1650069 ◽  
Author(s):  
M. Ismail ◽  
A. Y. Ellithi ◽  
A. EL-Depsy ◽  
O. A. Mohamedien
Keyword(s):  
Experimental Data ◽  
Coulomb Barrier ◽  
Alpha Decay ◽  
New Approach ◽  
Systematic Calculation ◽  
Barrier Penetration ◽  
Spherical Nuclei ◽  
New Formula ◽  
Penetration Probability ◽  
Good Agreement

A systematic calculation of alpha decay half-lives of 347 nuclei is considered in the framework of the Wentzel–Kramers–Brillouin (WKB) approximation using two formulas. A recently proposed barrier penetration formula, with some modified parameters, is used first. Second, a new analytic barrier penetration formula is derived by taking into account the centrifugal potential. A good agreement with experimental data is achieved especially for spherical nuclei. The new formula reproduces experimental alpha decay half-lives with a satisfying accuracy especially for penetration energies much lower than the Coulomb barrier.

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

ALPHA DECAY AND FISSION OF ALIGNED NUCLEI

1960 ◽  
Vol 38 (2) ◽  
pp. 290-314 ◽  
Author(s):  
N. R. Steenberg ◽  
R. C. Sharma
Keyword(s):  
Angular Distribution ◽  
High Temperature ◽  
Experimental Work ◽  
Low Temperatures ◽  
Alpha Decay ◽  
Alpha Particles ◽  
Nuclear Shape ◽  
Fission Fragments ◽  
Partial Waves ◽  
High Temperature Approximation

The theory of the angular distribution of alpha particles and of fission fragments from nuclei aligned at low temperatures is presented. Very explicit results are obtained in the high temperature approximation. These are directly dependent upon the branching which takes place to the various allowed partial waves. This branching is influenced by the nuclear shape, but it is shown that for this problem the effect of penetrating a spheroidal barrier is not critical. An application is made to the experimental work so far available and the result is reasonably satisfactory.

Fourier Transforms in Probability, Random Variables and Stochastic Processes

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Stochastic Processes ◽  
Probability Density ◽  
Density Function ◽  
Fourier Transforms ◽  
Random Variables ◽  
Cumulative Distribution ◽  
The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

scholarly journals A New Modeling Approach for the Probability Density Distribution Function of Wind power Fluctuation

2019 ◽  
Vol 11 (19) ◽  
pp. 5512 ◽  
Author(s):  
Lingzhi Wang ◽  
Jun Liu ◽  
Fucai Qian
Keyword(s):  
Distribution Function ◽  
Density Distribution ◽  
Probability Density ◽  
Wind Power ◽  
Rapid Development ◽  
Probability Density Distribution ◽  
Power Fluctuation ◽  
Function Model ◽  
New Model ◽  
Fireworks Algorithm

With the rapid development of grid-connected wind power, analysing and describing the probability density distribution characteristics of wind power fluctuation has always been a hot and difficult problem in the wind power field. In traditional methods, a single distribution function model is used to fit the probability density distribution of wind power output fluctuation; however, the results are unsatisfying. Therefore, a new distribution function model is proposed in this work for fitting the probability density distribution to replace a single distribution function model. In form, the new model includes only four parameters which make it easier to implement. Four statistical index models are used to evaluate the distribution function fits with the measured probability data. Simulations are designed to compare the new model with the Gaussian mixture model, and results illustrate the effectiveness and advantages of the newly developed model in fitting the wind power fluctuation probability density distribution. Besides, the fireworks algorithm is adopted for determining the optimal parameters in the distribution function model. The comparison experiments of the fireworks algorithm with the particle swarm optimization (PSO) algorithm and the genetic algorithm (GA) are carried out, which shows that the fireworks algorithm has faster convergence speed and higher accuracy than the two common intelligent algorithms, so it is useful for optimizing parameters in power systems.

Optimal temperature for alpha-decay half-lives with Yukawa proximity potential

2016 ◽  
Vol 25 (12) ◽  
pp. 1650109 ◽  
Author(s):  
S. S. Hosseini ◽  
H. Hassanabadi ◽  
S. Zarrinkamar
Keyword(s):  
Experimental Data ◽  
Excited State ◽  
Coulomb Potential ◽  
Alpha Decay ◽  
Nuclear Surface ◽  
Optimal Temperature ◽  
Proximity Potential ◽  
Excited System ◽  
Nuclear Temperature ◽  
Comparison Of The Results

The paper investigates the alpha-decay half-lives of some nuclei by modifying the Coulomb potential with Yukawa proximity potential for the excited state decays. A new relation is found for the width diffuseness of the nuclear surface [Formula: see text] and the sharp radii [Formula: see text] for the excited system. The parameters are fitted to the experimental data for the nuclear temperature in the range [Formula: see text] (MeV). A comparison of half-life indicates that the probability of decay increases with increasing nuclear temperature for the excited system. The comparison of the results with the existing experimental data is motivating.

scholarly journals Approximations to Risk Theory's F(x, t) by Means of the Gamma Distribution

1977 ◽  
Vol 9 (1-2) ◽  
pp. 213-218 ◽  
Author(s):  
Hilary L. Seal
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Gamma Distribution ◽  
Gamma Function ◽  
Function Approximation ◽  
Good Accuracy ◽  
Electronic Computer ◽  
Pocket Computer ◽  
Power Approximation ◽  
Aggregate Claims

It seems that there are people who are prepared to accept what the numerical analyst would regard as a shockingly poor approximation to F (x, t), the distribution function of aggregate claims in the interval of time (o, t), provided it can be quickly produced on a desk or pocket computer with the use of standard statistical tables. The so-called NP (Normal Power) approximation has acquired an undeserved reputation for accuracy among the various possibilities and we propose to show why it should be abandoned in favour of a simple gamma function approximation.Discounting encomiums on the NP method such as Bühlmann's (1974): “Everybody known to me who has worked with it has been surprised by its unexpectedly good accuracy”, we believe there are only three sources of original published material on the approximation, namely Kauppi et al (1969), Pesonen (1969) and Berger (1972). Only the last two authors calculated values of F(x, t) by the NP method and compared them with “true” four or five decimal values obtained by inverting the characteristic function of F(x, t) on an electronic computer.

scholarly journals Scattering of Alpha-particles by a Vibrational Nucleus

1962 ◽  
Vol 15 (2) ◽  
pp. 135 ◽  
Author(s):  
LJ Tassie
Keyword(s):  
Elastic Scattering ◽  
Plane Wave ◽  
Experimental Test ◽  
Born Approximation ◽  
Alpha Particles ◽  
Nuclear Surface ◽  
Vibrational States ◽  
Phonon Excitation ◽  
Plane Wave Born Approximation ◽  
Good Agreement

The elastic and inelastic scattering of cx-particles by a vibrational nucleus is calculated using plane-wave Born approximation. The excitation of both single-phonon and twophonon states is considered. The effect of the diffuseness of the nuclear surface is included. The result for elastic scattering and the excitation of the single-phonon 2+ and 3- states is in good agreement with experiment for .oNi. The approximations used are discussed, and it is suggested that excitation of 0+, 1-, and 5- states should provide the best experimental test of the theory of two-phonon excitation of nuclei. The energies of the vibrational states are also considered.

Interpretation of the photon: wave–particle duality

2012 ◽  
Vol 90 (9) ◽  
pp. 855-863
Author(s):  
Gilbert R. Hoy ◽  
Jos Odeurs
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Coherence Length ◽  
Probability Distribution Function ◽  
Three Dimensional ◽  
Space Time ◽  
Maximum Probability ◽  
Excited System ◽  
Wave Particle Duality

A simple model is provided to obtain the space–time probability-distribution function of a photon emitted without recoil by an excited system (atom, nucleus, …) in one dimension. A three-dimensional formulation is not needed for our discussion. A quantum mechanical calculation, using the Heitler method, is employed to obtain the solution. The space–time probability-distribution function is not the photon wavefunction. In fact, the area under the space–time probability-distribution function is time dependent. It obtains its final value only as t → ∞. The frequency composition of the photon is found and its time dependence determined to be in accord with the time–energy uncertainty principle. In the wave picture, the coherence length of a photon is found to be equal to the distance from the maximum probability-density position in the photon back toward the source to a position where the probability density has decreased to e–1 of its maximum value. The concept of the coherence length is applied to understand the exponential lifetime curve in the wave picture. This latter measurement is usually explained by saying, in the particle picture, that the photon can appear immediately after formation of the excited state or at a variety of later times according to an exponential probability distribution.

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