Radioactive decay defined

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 decay constant, X, defines the probability that a particular atom will decay within a given time (X = In 2/t1/2). The half-life (t1/2) describes a time interval after which N = NJ2. The observed counting rate or activity (A) is equal to XN. Another way to describe radioactive decay is in terms of the mean life (t) of a... 

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 ... 

Half-life is defined as the time required for a radioisotope to reduce its initial radioactivity (disintegration rate) to one-half (or 50%). The half-life is represented by the symbol, t a, and it is unique for a given radioisotope. The useful lifetimes of radiopharmaceuticals are usually determined by radioactive decay, which constantly decreases the amount of radioactivity present. The half-life is related to decay constant, X of a radioisotope (discussed in the subsequent section), as follows ... 

The fluorescence decay time is one of the most important characteristics of a fluorescent molecule because it defines the time window of observation of dynamic phenomena. As illustrated in Figure 3.2, no accurate information on the rate of phenomena occurring at time-scales shorter than about t/100 ( private life of the molecule) or longer than about 10t ( death of the molecule) can be obtained, whereas at intermediate times ( public life of the molecule) the time evolution of phenomena can be followed. It is interesting to note that a similar situation is found in the use of radioisotopes for dating the period (i.e. the time constant of the exponential radioactive decay) must be of the same order of magnitude as the age of the object to be dated (Figure 3.2). 

The concentration of the radioactive nuclide (reactant, such as Sm) decreases exponentially, which is referred to as radioactive decay. The concentration of the daughter nuclides (products, including Nd and He) grows, which is referred to as radiogenic growth. Note the difference between Equations l-47b and l-47c. In the former equation, the concentration of Nd at time t is expressed as a function of the initial Sm concentration. Hence, from the initial state, one can calculate how the Nd concentration would evolve. In the latter equation, the concentration of Nd at time t is expressed as a function of the Sm concentration also at time t. Let s now define time t as the present time. Then [ Nd] is related to the present amount of Sm, the age (time since Sm and Nd were fractionated), and the initial amount of Nd. Therefore, Equation l-47b represents forward calculation, and Equation l-47c represents an inverse problem to obtain either the age, or the initial concentration, or both. Equation l-47d assumes that there are no other ot-decay nuclides. However, U and Th are usually present in a rock or mineral, and their contribution to " He usually dominates and must be added to Equation l-47d. 

Because a diffusion profile does not end abruptly (except for some special cases), it is necessary to quantify the meaning of diffusion distance. To do so, examine Equation 3-40a. Define the distance at which the concentration is halfway between Co and to be the mid-distance of diffusion, Xmid- The concept of Xmid is similar to that of half-life ti/2 for radioactive decay. From the definition, Xmid can be solved from the following ... 

The number of atoms of Pb and Pb can be described as the number of atoms of each isotope present initially plus the number created in situ by radioactive decay. Tera and Wasserburg (1972, 1974) defined a concordia in parametric form where the x-coordinate is given by... 

Equations (5.1) define a direction vector at each point (t,y) of the n+1 dimensional space. Fig. 5.1 shows the field of such vectors for the radioactive decay model (5.2). Any function y(t), tangential to these vectors, satisfies (5.2) and is a solution of the differential equation. The family of such curves is the so called general solution. For (5.2) the general solution is given by... 

There are a multitude of papers concerning the age dating of rocks by mass spectrometry on the basis of different geochronological systems with well defined constants of radioactive decay. They include such systems as Rb-Sr, K-Ar, K-Ca, Re-Os, Nd-Sm, U-Pb, Th-Pb, Pb-Pb or Lu-Hf, which are discussed in detail in the literature.26 Therefore only a few relevant examples were briefly sketched here. 

Any radionuclide is characterised by its half-life r whose value is independent of the type of decay products that are created. Half-life is defined as the time required (from initial time t = 0) for the decomposition of half the atoms in the sample. The law of radioactive decay allows calculation of the number of atoms N left at time t in a population with N0 atoms initially. The integrated form of this law is given by the following equation ... 

Although we have found that for internal noise the Ito-Stratonovich dilemma is undecidable for lack of a precise A(t) there are cases in which the Ito equation seems the more appropriate option. As an example we take the decay process defined in IV.6 the M-equation is (V.1.7) and the average obeys the radioactive decay law (V.1.9). As the jumps are relatively small one may hope to describe the process by means of a Langevin equation. Following the Langevin approach we guess... 

Radioactive decay with emission of particles is a random process. It is impossible to predict with certainty when a radioactive event will occur. Therefore, a series of measurements made on a radioactive sample will result in a series of different count rates, but they will be centered around an average or mean value of counts per minute. Table 1.1 contains such a series of count rates obtained with a scintillation counter on a single radioactive sample. A similar table could be prepared for other biochemical measurements, including the rate of an enzyme-catalyzed reaction or the protein concentration of a solution as determined by the Bradford method. The arithmetic average or mean of the numbers is calculated by totaling all the experimental values observed for a sample (the counting rates, the velocity of the reaction, or protein concentration) and dividing the total by the number of times the measurement was made. The mean is defined by Equation 1.1. 

This is similar to a first-order reaction in chemical kinetics and follows the same law as radioactive decay. The rate constant kv defined in this manner is the natural radiative rate constant which also defines the natural radiative lifetime... 

Many of these effects of radioactive decay can be treated quantitatively using G values. Historically, the G value was defined as the number of molecules or species decomposed or formed per 100 eV of absorbed energy. A newer (SI) definition of the G value is the number of moles of molecules or species formed or decomposed per Joule of energy absorbed. (Note that 1 mol/J = 9.76 x 106 molecules/100 eV.) The G values depend on the radiation and the medium being irradiated and its physical state. Table 19.1 shows some typical G values for the irradiation of neutral liquid water. 

Why do some nuclei undergo radioactive decay while others do not Why, for instance, does a carbon-24 nucleus, with six protons and eight neutrons, spontaneously emit a /3 particle, whereas a carbon-23 nucleus, with six protons and seven neutrons, is stable indefinitely Before answering these questions, it s important to define what we mean by "stable." In the context of nuclear chemistry, we ll use the word stable to refer to isotopes whose half-lives can be measured, even if that half-life is only a fraction of a second. We ll call those isotopes that decay too rapidly for their half-lives to be measured unstable, and those isotopes that do not undergo radioactive decay nonradioactive, or stable indefinitely. 

For a steady-state system, a time-dependent model is used because of the irregular shape of the atmospheric 14C02 record. This model accounts for radioactive decay of the 14C since 1950 explicitly, and it requires that we compare measured radiocarbon to a standard with a radiocarbon value that stays constant over time (Aabs). For ease, we define F here as ASN/Aabs [see Eq. (A1.4)] for samples measured since 1950 F equals A14C/1000 + 1. For a reservoir at steady state, the balance of radiocarbon entering and leaving the reservoir in year t is given by... 

From day 6 onwards, the slope of the curve corresponds to the effective half-life of 131I on herbage, namely 5 d. Assuming that this continues indefinitely, the area under curve A in Fig. 3.6 is 1.4 m2 d l-1. This is equivalent to the transfer factor km, defined by equation (2.12). Values of Fm for 131I and 137Cs are about the same, but the radioactive decay of 131I reduces km compared with that for137Cs (Table 2.19). Also shown in Fig. 3.6 are values of C/ as deduced from measurements near... 

Decay rate is way to quantify radioactive decay and is equal to the number of radioactive decays or disintegrations occurring per unit time. The official SI unit of decay rate is the becquerel (Bq) defined to be ... 

The rate of radioactive decay is by convention expressed as the half-life, T1/2, defined as the time span during which a given concentration of the radioactive element atoms decays to half their initial value. T1/2 of tritium is 12.3 years. Thus, after 12.3 years one-half the initial concentration of tritium atoms is left, after 24.6 years only one-quarter is left, and so on. A radioactive decay curve of tritium is given in Fig. 10.1. Using the decay curve it is possible to determine, for example, how many years it takes for a given amount of tritium to decay to 20% of the initial amount. The answer, obtained from Fig. 10.1, is 29 years. Similarly, one can determine what percentage of an initial amount of tritium will be left after 20 years. The answer is 32% (read from Fig. 10.1). 

Fig. 12.1 A radioactive decay curve of 36C1. P36 is the percentage of atmospheric 36C1 left in the sample. The time axis is expressed in half-lives and in 105-year units. The marked point of 56% P36, taken from the Lachlan case study, reveals water with an age of about 290,000 years. The mode of defining the P36 is explained in later sections.

Because of the differences in the behavior of the particles and rays produced by radioactive decay, both the energy dose of the radiation and its effectiveness in causing biological damage must be taken into account. The rem (which is short for roentgen equivalent for man) is defined as follows ... 

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