Nuclear Decay and Radioactive Series

Nuclear decay occurs whenever a nucleus is in an energy state which is not the lowest possible for its "nucleon number" (A; a nucleon is a proton or a neutron). This state may occur naturally (which essentially means that it was created in that state in an earlier generation star) or by artificial means (neutron or photon irradiation). The lowest state can be computed using a "shell model" which uses techniques not unlike the Hartree approximation; in particular, the energy level depends heavily on the angular momenta and spins of the nucleons. There are several modes of decay:

  1. alpha (which we have already seen in section A);

  2. "beta" (electron or positron emission);

  3. "gamma" (high energy photon emission; a nuclear process as opposed to x-rays, which are the result of atomic electron transitions for very heavy elements)

  4. and electron (or positron) capture (the opposite of beta decay).

These decays are called by the common name of "radioactivity". Radioactive "isotopes" (atoms with the same Z but different A and N) are designated by "A element Z", as in 235U 92.

As we saw in section A, an alpha particle is a 4He++: a Helium nucleus. Alpha decay is most common for elements with Z > 82, since it provides the greatest energy loss per nucleon. In alpha decay, the following nuclear parameters change:

Z -> Z - 2 A -> A - 4 N -> N - 2

Beta decay is the "transformation" of a neutron into a proton, with the emission of an electron for charge conservation, and an antineutrino (see section E below) for energy and momentum conservation. It occurs in those situations in which alpha decay would leave the nucleus less stable than it was before. In Beta decay, the following parameters change:

Z -> Z + 1 A -> A N -> N - 1

Another form of Beta decay occurs in which a proton changes into a neutron plus a positron (see Section E below) and a neutrino, with parameter changes:

Z -> Z - 1 A -> A N -> N + 1

The capture processes are the opposite of the emission processes.

Gamma decay occurs because there is a conservation rule for nucleii which is violated by some of the other decay modes. We define the "nuclear parity" to be "even" if the (very complicated) function y which describes the nucleus has the property y(x) = y(-x), and "odd" if y(x) = - y(-x) (like even and odd functions in algebra). Nuclear parity "is conserved" (the parity of the nucleus function cannot change), and beta decay violates this for some nucleii. Therefore some nucleii cannot beta decay; they gamma decay instead. In addition to conservation of parity, nuclear reactions must also conserve energy, charge, momentum, and angular momentum, and must lower the energy of the nucleus.

Typically, one unstable nucleus will decay into another unstable nucleus, over and over in a "decay series" until an ultra-stable nucleus (usually 206, 207 or 208Pb 82) is reached as the end product. Three such series occur naturally, one beginning with 232Th 90, and the others beginning with 238U 92 and 235U 92. The Thorium Series is

The "branches" in the decay series occur when two decay modes are possible with (usually) differing probabilities. We can never tell for certain when any given nucleus will decay, and if a branch is possible, which branch it will take when it does decay. For this reason, radioactive decay rates are based on large samples (numbers on the order of Avogadro's number; see the next chapter).

In a large population of unstable nucleii, the decays occur apparently at random, but the overall decay rate is described by an empirical exponential function

D = D 0 2 - t / t.

Here D 0 is the initial amount , D is the amount left at time t, and the base 2 is used since t is a "half-life" (the time it takes for D to be 1/2 of D 0). The half life is proportional to the stability of the excited nucleus: more stable nucleii have longer half lives, etc. It is somewhat analogous to the time constant in an RC circuit: it is a characteristic quantity for a given isotope. D may be measured in units of mass, but is often measured in Curies (Ci, corresponding to about 1g of Ra, or about 3.7 x 10 10 decays / s).

Problems

The next section is on the biological effects of radiation.



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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