Decay statistical nature

Statistical nature of some processes, e.g., radioactive decay. There is a probability that an atom of a radioactive isotope will decay in the next 10 s, and this is as much information as one can report on this matter. The probability can be calculated, but it is still a probability, never a certainty. 

Even if the experimental design and execution are perfect so that the determinant error is eliminated in experiments involving radioactivity there is always a random error due to the statistical nature of radioactive decay. Each radioactive atom has a certain probability of decay within any one time interval. Consequently, since this probability allows unlikely processes to occur occasionally and likely processes not to occur in any particular time interval, the number of decays may be more or less than the number in another similar... 

The statistical nature of radioactive decay also leads to an uneven distribution of decays in time which is important when handling dead-time corrections and discussing required system time resolution. Let us first assume that a decay has occurred at time t = 0. What is then the differential probability that the next decay will take place within a short time interval, dr, after a time interval t has passed Two independ t processes must then occur in series. No decay may take place within the time interval from 0 to r, probability P 0),... 

On account of the statistical nature of radioactive decay it is advisable to perform each measurement three times. 

As a measure for the mixture homogeneity the relative standard deviation of the measured count rates can be used. It is important to subtract the variations due to the statistical nature of radioactive decay, which would otherwise contribute to the measured variance. It has to be noted that the determination of homogeneity is not unambiguous. The variations in tracer concentration of samples depend on the sample size and on the microstructure of the material. 

Fluorescence decay in a population of excited-state molecules is a random process, so that the arrival time of the first photon is also random, although it is more likely to occur at the peak of the fluorescence decay than it is along the latter s tail. Because of this inherent statistical nature, over many excitation cycles one gradually builds us a statistical profile of the likelihood of emission versus time that corresponds to the decay profile. For a large enough number of counts, the uncertainty in the number of counts in a particular channel is equal to the count number s square root, which gives us the weighting factors to be used in the least-squares analysis of the data. 

While it is true to say that all scientific measurements are estimates of some unattainable true measurement, this is particularly true of radioactivity measurements because of the statistical nature of radioactive decay. Consider a collection of unstable atoms. We can be certain that all wiU eventually decay. We can expect that at any point in time the rate of decay will be that given by Equation (5.1). However, if we take any particular atom we can never know exactly when it will decay. It follows that we can never know exactly how many atoms will decay within our measurement period. Our measurement can, therefore, only be an estimate of the expected decay rate. If we were to make further measurements, these would provide more, slightly different, estimates. This fundamental uncertainty in the quantity we wish to measure, the decay rate, underlies ah radioactivity measurements and is in addition to the usual uncertainties (random and systematic) imposed by the measurement process itself. 

The nature of the radiative decay in the resonance and in the statistical limits was considered by Berry and Jortner,7 who examined the interference effects in the radiative decay of coherently excited states. Quantum beat signals can be observed and used to analyze close-lying molecular... 

Expression and Interpretation of Results. Archaeological interpretation of a radiocarbon age may depend critically on the error associated with that age. Errors are commonly expressed as a variance range attached to the central number (e.g., 2250 80 years). The 80 years in this example may correspond to the random error for a single analytical step. Both decay and direct-atom counting are statistical in nature, and lead to errors that vary as the square root of the number of counts. The error may also be expressed as the overall random experimental error (the sum of individual errors.) Overall random error can be determined only by analyzing replicate samples. 

If the spontaneous emission of radiation of the appropriate energy is the only pathway for a return to the initial state, the average statistical time that the molecule spends in the excited state is called the natural radiative lifetime. For an individual molecule the probability of emission is time-independent and the total intensity of emission depends on the number of molecules in the excited state. In a system with a large number of particles, the rate of decay follows a first-order rate law and can be expressed as... 

Radioactive decay is a statistical process, there being nothing in any nucleus that allows us to predict when it will decay. The probability of decay in a given time interval is the only thing that can be determined, and this appears to be entirely constant in time and (except in the case of electron capture) unaffected by temperature, pressure or the chemical state of an atom. The probability is normally expressed as a half-life, the time taken for half of a sample to decay. Half-lives can vary from a fraction of a second to billions of years. Some naturally occurring radioactive elements on Earth have very long half-lives and are effectively left over from the synthesis of the elements before... 

TTHE MOST IMPORTANT FORCES ACTING BETWEEN MEMBRANE SURFACES are van der Waals, electrostatic, and hydration. The first two forces are explained by the Deijaguin-Landau-Verwey-Overbeek (DLVO) theory (I) the existence of the hydration force was anticipated before it was measured (2). The van der Waals force is always attractive and displays a power law distance dependence, whereas the electrostatic and hydration forces are repulsive and exponentially decay with distance. The electrostatic force describes the interaction between charged membrane surfaces when the separation between surfaces is above 10 molecular solvent diameters. The hydration force acts between charged and uncharged membrane surfaces and at distances below 10 molecular solvent diameters its value dominates the values of van der Waals and electrostatic forces (3). The term hydration reflects the belief that the force is due to the structure of water between the surfaces. Electrostatic and hydration forces are similar in some respects both are exponential and repulsive and their theoretical description involves coupling electrostatic concepts and ideas borrowed from statistical mechanics. Although the nature of the electrostatic force is solidly established, this is not the case for the hydration force. To illustrate the role the electrostatic... 

The results of radiation measurements are, in most cases, expressed as the number of counts recorded in a scaler. These counts indicate that particles have interacted with a detector and produced a pulse that has been recorded. The particles, in turn, have been produced either by the decay of a radioisotope or as a result of a nuclear reaction. In either case, the emission of the particle is statistical in nature and follows the Poisson distribution. However, as indicated in Sec. 2.9, if the average of the number of counts involved is more than about 20, the Poisson approaches the Gaussian distribution. For this reason, the... 

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