Radioactive nuclei, decay rates

A radioactive nucleus decays by a first-order process, so that (20-1), (20-2), and (20-3) apply. The stability of the nucleus with respect to spontaneous decay may be indicated by its first-order rate constant, k, or by the half-life, ti/2-... 

Figure 17.8 Schematic of neutron activation. When a neutron interacts with the target nucleus an isotope forms that can be unstable. If so, it will almost instantaneously de-excite into a more stable configuration through emission of y-rays. Then, this new radioactive nucleus decays by emission of an electron and one or more characteristic y-rays, at a rate according to the half-life of this nucleus. Illustration with Ag atom.

The radioactive nucleus decays spontaneously at a rate which is proportional to the number of active (unstable) nuclei... 

Another kind of particle and another kind of interaction were discovered from a detailed study of beta radioactivity in which electrons with a continuous spectrum of energies are emitted by an unstable nucleus. The corresponding interactions could be viewed as being due to the virtual transmutation of a neutron into a proton, an electron, and a new neutral particle of vanishing mass called the neutrino. The theory provided such a successful systematization of beta decay rate data for several nuclei that the existence of the neutrino was well established more than 20 years before its experimental discovery. The beta decay interaction was very weak even compared to the electron-photon interaction. 

Conversion of a radioactive parent to a single stable daughter, as occurs for is the simplest form of nuclear decay. Although it is impossible to determine when an individual radioactive nucleus will convert, decay rates become predictable for large populations of... 

The SNF portion of HLW can be understood by chemists who see in it nearly every element on the periodic chart of the elements. After a 235u nucleus undergoes fission and releases its excess nuclear binding energy, it leaves a pair of new atoms. These fission products are like newly born forms of the elements that are already well known and, like newborns, are unstable until they mature. There are about 1000 isotopes of about 100 different elements in SNF, and most are radioactive. They decay into stable elements at different rates, giving off alpha, beta, and gamma emissions. It will take about 7000 years until the SNF will be only as radioactive as the rocks and minerals that make up our planet. 

The probability that a radioactive nucleus will decay in a given time is a constant, independent of temperature, pressure, or the decay of other neighboring nuclei. The disintegrations of individual nuclei are statistically independent events and are subject to random fluctuations. In a large number of nuclei, however, the fluctuations average out, and the fraction that decays in unit time is a constant and is numerically equal to the probability that a single nuclei will decay in that time. This rate of radioactive decay is known as the decay constant X, with dimensions of reciprocal time. 

A radioactive nucleus which emits a particle to become transformed to another nucleus is described as decaying to that nucleus. Such a radioactive event is called radioactive decay. Radionuclides decay at different rates. Some can decay in millionths of a second, others take millions of years. Decay is independent of all the variables which affect chemical reactions such as temperature, pressure, and concentration. This poses particular difficulty with regard to the disposal of nuclear wastes. The rate of radioactive decay is characterized by the loss of a constant percent per unit time, not a constant number of moles per unit time. We therefore characterize the decay rate by specifying the time required for 50 percent of the original material to decay. This period of time is called the half-life, given the symbol, tj/j- The constant percent change means that 50 percent will be lost during the first half-life, 50 percent of what is left after the first half-life will decay over the second half-life, etc. 

The probabflity of a nucleus decaying is proportional to the number of nuclei present, and all radioactivity processes follow first-order rate laws. The rates are generally independent of the temperature and the chemical environment that surrounds the nuclei. Because the rate is proportional to the number of nuclei, we can write... 

One of the most important properties of a radioactive nuclide is its lifetime. At present it is not possible to predict theoretically when any particular nucleus in a sample will decay. However, the number of nuclides in a sizeable sample that will decompose in a given time can be measured, and it is found that this rate of decay is characteristic of a given isotope. In fact, the rate of decay of an isotope is constant and unvarying. That is, if a fraction of a radioactive nuclide decays in a certain time interval f, then the same fraction of the remainder will decay in another increment of time f, irrespective of external conditions. Nuclear reactions are not affected by outside influences such as temperature and pressure and it is not possible to significantly alter the constant rate of radioactive decay. For example, radioactive strontium-90, an important... 

Solution (a) Analyze and Plan We are asked to calculate a half-life, fj/2, based on data that teU us how much of a radioactive nucleus has decayed in a given period of time (Nq = 1.000 g, A4 = 0.953 g, and t = 2.00 yr). We do this by first calculating the rate constant for the decay, k, then using that to compute fj/2. 

You can obtain the decay constant for a radioactive nucleus by counting the nuclear disintegrations over a period of time. The original definition of the curie (now 3.7 X 10 disintegrations per second) was the activity or decay rate of 1.0 g of radium-226. You can use this with the rate equation to obtain the decay constant of radium-226. Radium-226 has a molar mass of 226 g. A 1.0-g sample of radium-226 contains the following number of nuclei ... 

Although you might be able to obtain the half-life of a radioactive nucleus by direct observation in some cases, this is impossible for many nuclei because they decay too quickly or too slowly. Uranium-238, for example, has a half-life of 4.51 billion years, much too long to be directly observed The usual method of determining the half-life is by measuring decay rates and relating them to half-lives. 

Radioactive decay of nuclei is a first-order reaction decay rate (activity A) is therefore dependent on the concentration (content) of the radionuclide and is the product of this concentration (more precisely, the number of atoms of radionuclide N) and the decay constant X (in s" ) A =-(dN/dt) = X.N. The basic unit of activity, according to the System International (SI system) is the Bq (becquerel). One Bq (in s ) is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. Previously, the frequently used unit was the Ci (curie) defined as 3.7 x 10 decays per second. For conversion, the following relationship can be used 1 Ci = 3.7 x lO Bq. Number of radionuclide atoms transformed in time t is f T=NQ.e", where Nq is the initial number of atoms of the radionuclide at the time t=0. During conversion, the number of radioactive atoms of the radioactive nuclide is continuously decreasing. Combining both equations we get the relation expressing the dependence of activity on time A = -(dN/dt) The... 

Methods for influencing the radioactive decay rate have been sought from early years of Nuclear physics. Nuclear transmutation (i.e. change in the nuclear charge) induced by nuclear reactions are often accompanied by a redistribution of the electrons, muons (mesons in the hadronic atoms) around the final transmuted nucleus. Muonic atoms have always been useful tools for nuclear spectroscopy employing atomic-physics techniques. Muonic atoms also play an important role... 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

The basic concepts of nuclear structure and isotopes are explained Appendix 2. This section derives the mathematical equation for the rate of radioactive decay of any unstable nucleus, in terms of its half life. 

The rate of decay of an unstable parent nucleus at any time t is proportional to the number (N) of atoms left (Faure, 1986 38). In other words, the rate at which the number of radioactive nuclei decline is proportional to the number left at that time. Expressed mathematically, this becomes ... 

Equation (9.6) is the basic equation describing the decay of all radioactive particles, and, when plotted out, gives the familiar exponential decay curve. The parameter X is characteristic of the parent nucleus, but is not the most readily visualized measure of the rate of radioactive decay. This is normally expressed as the half life (7/ 2). which is defined as the time taken for half the original amount of the radioactive parent to decay. Substituting N = Na/2 into the Equation (9.6) gives ... 

In essence, NAA involves converting some atoms of the elements within a sample into artificial radioactive isotopes by irradiation with neutrons. The radioactive isotopes so formed then decay to form stable isotopes at a rate which depends on their half-life. Measurement of the decay allows the identification of the nature and concentration of the original elements in the sample. If analysis is to be quantitative, a series of standard specimens which resemble the composition of the archaeological artifact as closely as possible are required. NAA differs from other spectroscopic methods considered in earlier chapters because it involves reorganization of the nucleus, and subsequent changes between energy levels within the nucleus, rather than between the electronic energy levels. 

Apart from these three facts, nuclear astrophysicists take pains to point out that the rate at which the luminosities of SNla events decline, once beyond the maximum, is commensurable with the decay of radioactive cobalt-56, son of nickel-56, atomic nucleus of noble lineage as we know. This is a common factor with gravitational collapse supernovas. SNla light curves are explained through the hypothesis that half a solar mass of nickel-56 is produced when one of these white dwarfs explodes. 

For use in geochronology, the decay constant of a radioactive nuclide must be constant and must be accurately known. For a-decay and most (3-decays, the decay constant does not depend on the chemical environment, temperature, or pressure. However, for one mode of 3-decay, the electron capture (capture of K-shell electrons), the decay "constant" may vary slightly from compound to compound, or with temperature and pressure. This is because the K-shell (the innermost shell) electrons may be affected by the local chemical environment, leading to variation in the rate of electron capture into the nucleus. The effect is typically small. For example, for Be, which has a small number of electrons and hence the K-shell is easily affected by chemical environments, Huh (1999) showed that the decay constant may vary by about 1.5% relative (Figure l-4b). Among decay systems with geochronological applications, the branch decay constant of °K to °Ar may vary very slightly (<1% relative). 

One radioactive decay pathway, electron capture, has been found to have a minor dependence on external conditions of relevance to cosmochemistry. The rate at which an electron in the cloud surrounding the nucleus is captured by a proton to make a neutron is... 

The spontaneous disintegration of a nucleus is a first-order kinetic process. That is, the rate of radioactive decay of TV atoms (—dN/dt, the change of TV with time, t) is proportional to the number of radioactive atoms present (Equation 6.4). 

For very many years, the alchemist s dream of changing base metals into gold was ridiculed even by the most reputable of scientists. Although it was known that the nuclei of certain atoms undergo alteration in the course of natural radioactive decay, researchers inability to exercise any control over the nature or rate of these spontaneous decompositions probably did much to foster the belief that the nucleus of the atom was inviolate. However, in the year 1919 the English physicist Ernest Rutherford accomplished the first transmutation of an element, and this notable discovery was quickly followed by other equally significant developments. 

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