A radioactive decay is described as branching when one parent element decom­poses to two daughter nuclides. This type of decay can be characterized by two decay constants and half-lives as follows:

B

A

2 B2

where A is the parent nuclide, and B2 are the daughter nuclides, and Ai and A2 are the decay constants for the production of B1 and B2, respectively. Examples of such decay are the decomposition of the Pb isotope into Po and Tl, the decay of 64Cu isotope to 64Zn and 64Ni, and the disintegration of the 40K isotope into 40Ca and 40Ar isotopes.

Since during branching decay, the quantity of the parent element decreases via two independent processes, the rate of decay of the parent element can be defined by the sum of the two decay constants:

dN

— — = (Ai 1 A2)N dt

From here,

By the integration of Eq. (4.16): ln N = —(Ai 1 A 2)t 1 constant

Assuming that at t = 0, N = N0:

N = N0e2(Al+A2)t (4.18)

Equation (4.18) is similar to the kinetics of the simple radioactive decay (Eq. (4.8)), except that the sum of the individual constants is used as the decay con­stant. In those cases, when the daughter elements formed through different decay mechanisms or the energy of the emitted radiation is sufficiently different, the values of the decay constants can be determined separately. The proportion of the decay constants will determine the relative quantity of the daughter nuclides formed.

In most cases, however, both daughter elements are formed via beta decay, the spectra of which is continuous (see Section 4.4.2), and the decays are very difficult or impossible to separate. In this case, the ratio of the quantity of the daughter ele­ments can be calculated as follows.

The sum of the quantities of the two daughter elements is equal to the quantity of the decomposed parent element at any time:

B1 + B2 = N0 — N = N0(1 — e-(A1+A2)t) (4.19)

The rate of the formation of the daughter elements is:

^ = A1N = A1N0 e-(A1+A2)t dt

^ = A2N = A2N0 e-(A1+A2)t dt

By integrating Eqs. (4.20) and (4.21) from t = 0 to 00:

we obtain:

, = A1

!1 = ‘

A2

B2 — AT+X N0

The ratio of Eqs. (4.24) and (4.25) is:

Therefore, in branching decay, the ratio of the quantities of the daughter ele­ments is equal to the ratio of the decay constants. By determining the quantities of the daughter elements, the ratio of the decay constants can be calculated.

Equations (4.20) and (4.21) have been integrated from t = 0 to oo, but the same results are obtained by the integration over any time interval.