Bateman equation

In-situ decay, no initial daughter nuclides, no melting (Bateman Equation)... 

Example Problem Consider the decay of a 1 p,Ci sample of pure 222Rn (t /2 = 3.82 d). Use the Bateman equations to estimate the activity of its daughters (218Po, 214Pb, 214Bi, and 214Po) after a decay time of 4 h. 

To obtain the amount A, (t). we consider a time interval from t to t + dt where t < t. During this interval dt, the amount of species 1 produced is P dt. The ultimate decay of this amount P dt of species 1 over the interval t — t results in a net amount dNi(t, t) obtained by applying the Bateman equation (2.17) ... 

A general equation for the amount of any nuclide present at a time t after removal from the reactor can be derived by using the Bateman equation (2.17). Consider a time variable t, such that when 0 < t < T the number of atoms of the first member of the chain produced during any interval dt is P dt. The relevant time scale is shown below ... 

Example 3.11 Apply the Bateman equation for the second and third isotope in a series. 

The general solution to the case with many successive decays is usually referred to as the Bateman equations (H. Bateman 1910) ... 

Case (Hi) Combined iruluced transformation and radioactive decay. These combined effects can be taken into account by using the Bateman equations with some modifications, as developed by Rubinson. 

Calculations for longer decay chains under some conditions can be simplified by assuming that short-lived daughters and long-lived parents have the same disintegration rate. In some complex chains, the Bateman equation (in the same form as Eq. (2.8), but with terms added to describe further decays) can be used to determine the ingrowth and decay pattern of three or four successive radionuclides (Evans 1955). 

The solution of the above system of equations can be found, e.g., in Vertes and Kiss (1987), Ehmann and Vance (1991) as well as several authors of this handbook refer to the result summarized by O Eqs. (7.42)-(7.43) as the Bateman equation (Bateman 1910, see also in Choppin and Rydberg 1980). Other authors refer to the whole series of equations shown in O Eq. (7.40) as Bateman equations.) For the nth member of the series (n = 1, 2, 3,...) one gets... 

For more generalized forms of the above Bateman equations including activation by neutron capture see Eqs. (1.29)-(1.30) in Chap. 38, Vol. 4, cited from Rubinson (1949). 

See O Chap. 7 of Vol. 1 for information on the rates of genetically related radioactive materials and, in particular, the Bateman equations describing members of arbitrarily long chains. Note that each set of equations describes the components of a chain defined for particular branches. Different branches require different set of equations. These points are illustrated in the following sections. 

The decay of Ac to Th is an example of transient equilibrium. (See Fig. 13.3 for half-lives and branching ratios.) An initially pure sample of Ac will have increasing amounts of Th until the rate of decay of Th is nearly the same as the rate of decay of Ac to it. If the Th were fed at a constant rate, it would reach 99.9% of that rate in 9.966 x 18.7 days 186.6 days. However, during that time the Ac would decay by 1.61%. Therefore the proper Bateman equation must be used to include both growth of Th and decay of Ac. The maximum in the decay rate of Th occurs when it is equal to the production rate, namely, 0.9858 of the initial rate of decay of Ac to Th at 3,930 h (164 days). 

One can consider the terms in the Bateman equations to have three parts a product of A s in the numerator, a product of differences in A s in the denominator, and an exponential factor. For branched decay of component i, the 1,- in the numerator is the partial decay constant, and the 1/s in the denominator and the exponential for i are the total decay constants. Another way to view the use of partial decay constants in the numerator is to multiply the entire equation by the fraction for that particular branch. This is not a simplification, because chains with different branches have different numerators and some different exponential factors. 

The calculation was based on a simplified form of the Bateman equation - see Eq. (7.42) in Chap. 7, Vol. 1 - for closed system radioactive decay to describe the °Po content of degassed lavas as a function of time. Using appropriate initial conditions for solving O Eq. (7.36) in Chap. 7, Vol. 1, one obtains the following relationship for the activities ... 

The exact solution to the corresponding differential equations, the Bateman equations, can be written as... 

In chronometry, the age of the sample is defined not in terms of the decay of a parent nuclide, but rather as the in-growth of a daughter activity. Radionuclides that are linked to one another by the processes of radioactive decay have relative concentrations that can be calculated with the Bateman equations, which express the simple laws of radioactive decay and ingrowth. If there exists a time at which all the descendant radionuclides have been removed from the mother material, that time can be determined through the measurement of the relative concentrations of the mother and daughter nuclides at a later time. The time interval between the purification of the sample and the subsequent analysis of the sample is defined as the age of the material at the analysis time. The technique does not apply when the half-life of the daughter nuclide involved in the determination is significantly shorter than the elapsed time. 

The Bateman equation was used at two different times for calculation of the initial °Sr/ V activities. That is to say, total beta activity, i.e. °Sr activity and its daughter was measured at two different times, one along the first day after the separation process and another when the secular equilibrium was reached. Main advantages of this method are the little amount of resin used and its long durability (up to 30 injections) achieving an MDA of 0.008 Bq °Sr. 

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