Radioactive Decay

The spontaneous emission of radiation from a nucleus is called radioactivity. When a nuclide changes to another element with the emission of radiation, it is called transmutation. There are three types of radiation that we will discuss. (For more on radioactivity and its history, click on http://www.accessexcellence.org/AE/AEC/CC/radioactivity.html)

Alpha particles (a) are simply helium nuclei (2 protons and 2 neutrons) and have a positive charge. a particles have relatively low penetrating power - they can be stopped by skin or by a piece of paper. They can have important biological effects on soft mucous tissue, such as inside of the lungs. Radium is a naturally occurring a emitter which decays to Radon with the emission of an a particle. We write the reaction thus:

Note here that an alpha particle is commonly written as a helium nuclide.

In this type of reaction, we define a parent (the original isotope) and a daughter (what is left after the decay). Here, radium is the parent and radon is the daughter. The mass of the parent is always greater than the mass of
the daughter and the mass of the
a particle. This mass difference is kinetic energy from the recoiling daughter and a particle and is called disintegration energy, Q. Q = (Mp - Md - Ma)c2. The computation is fairly straight forward.
For example, what is the disintegration energy for a uranium-thorium decay chain which emits an
a particle?

The decay is given by:

U-232 has a mass 232.037131 u and Th-228 has a mass 228.028716 u. The a has a mass of 4.002602 units. Thus, the mass lost is (232.037131 - 220.028716 - 4.002602 u) = 0.005813 u. This equates to an energy of (0.005813 u)(931.5 MeV/u) = 5.4 MeV.

Beta particles (b -) are electrons, and Beta decay occurs when a nuclide undergoes transmutation with the emission of an electron. It must be noted here that a b - particle is indistinguishable from an orbital electron, but it is in fact an electron that is generated within the nucleus by a neutron that decays to a proton, an electron, and a neutrino. A b - has more energy than an a particle, and can penetrate skin but can be stopped by about 3 mm of aluminum and other substances.

Another particle - the neutrino (n).  While conducting experiments in the late 1920s and early 1930s, it was found that the kinetic energy of a b - particle had significantly less energy in a decay reaction than they should have. Some physicists, including Niels Bohr, were ready to postulate that these types of reactions violated the law of conservation of energy. Wolfgang Pauli was the first to propose that a new particle , the neutrino, was also being generated that was virtually impossible to detect. It was thought that this particle was using all the energy and momentum that could not be accounted for in the b - decay reaction. The neutrino was postulated to have no
charge and zero rest mass, similar to the photon. In 1934,
Enrico Fermi (who named the neutrino) worked out a theory for b - decay that included the neutrino and this became generally accepted. It wasn't until the mid 50s that experimentation provided further evidence for the neutrino.

Antiparticles - Very simply, these are particles with characteristics similar to given particles but opposite in a given characteristic such as spin or charge. For example, an electron with a positive charge is called a positron and has the symbol b +. A neutrino has a particle with opposite spin called an antineutrino. Thus, we write the b - decay equation as

where P is the parent nuclide and D is the daughter and an electron and anti neutrino are also produced. As an example, Carbon 14 decays as

An example of positron decay occurs with Neon 19:

Gamma particles (g). A gamma ray is a photon with a very high energy. It has the capability to pass through several inches of steel and requires significant shielding design in nuclear reactors. The g has no charge and there is no transmutation of an element as a result of g decay. g are emitted when a nucleus drops from an excited state to a lower energy state. This can happen as the result of a high energy collision, or more commonly a nuclide will remain in an excited state following a previous decay.

Conservation Laws - these three types of decay exhibit all of the classic conservation laws that we have studied: Energy, momentum, electric charge are all conserved. Additionally, mass number is also conserved. This is called the Law of Conservation of Nucleon Number, and very simply states that the total number of nucleons is constant in any process, even though one nucleon may change into another type.

Half Life is the amount of time required for a substance to decay to one half of its original amount. Half lives can be measured in nanoseconds or in thousands of years, depending upon the material. Half life is constant for a given nuclide and is independent of the original amount of the material or external conditions. For example, if a material had a half life of 10 minutes, and originally we started with 8,000 grams of the material, every 10 minutes we would have 1/2 of the amount from the previous 10 minutes. We could develop a table that looks like this:

Time

Amount

0 minutes

8,000 grams

10 minutes

4,000 grams

20 minutes

2,000 grams

30 minutes

1,000 grams

40 minutes

500 grams

50 minutes

250 grams

60 minutes

125 grams

70 minutes

63 grams

The decay curve for this function looks like the graph at the right. Here, our time constant is .0693. We'll show to compute this a little later. Eventually we will reach a point where there is not any of the original substance left. In
practicality, for most substances we can say that the material is approximately gone after about 7 half lives. This is because after that time span, what remains is a very small portion of the original. Mathematically, we say that
the number of decays
DN that occur in a time interval Dt may be represented as a proportion to the original  amount, N, and the decay constant, l. l is different for different nuclides. The larger l is, the greater the rate of decay. We represent this with the equation DN = -lNDt. If we solve for N, we get a more generalized form of the equation which is useful:

N = Noe-lt

where No is the original amount of the material we had, t is the amount of time that has elapsed, and N is the amount remaining. The number of decays per second, or DN/Dt is called the activity of the substance. This number will decrease over time since there will be fewer nuclides left to decay. As an example: Carbon 14 is used for dating formerly living material since the ratio of Carbon 14 to Carbon 12 can be established accurately in living material and Carbon 14 is radioactive with a half life of 5730 years. If a plant had 12 grams of Carbon 14 when it died, how much C 14 remains after 15,000 years?

First, we need to find the time constant, l. From our equation above, N/No = 1/2 (half is decayed away in 1 time constant), .5 = e-lt or 2 = elt.
Taking the natural log of both sides, ln 2 = ln (e
lt) or .693 = l t. t1/2 = 5730 years

so l = .693/5730 years = 1.21 x 10-4 yrs-1.

Thus, after 15,000 years we have N = (12 g) e -(1.21 x 10-4 yrs-1 )(15,000yrs) = 1.95 grams remaining.

Often, a parent nuclide will decay to a daughter nuclide which will in turn decay to another daughter and continue
on in this fashion until a stable nuclide is reached. We call such a reaction a decay series. The chart at the right depicts the decay chain for 238U. The chart shows each nuclide, its form of decay, and the half life of the decay. For example, 238U decays to 234Th by
a emission with a half life of 4.5 billion years. This in turn b decays to 234Pa in 24 days which in turn b decays to 234U in 6.7 hrs. Note that some of these decays take years, while others occur in days, hours, minutes, and even tenths of a millisecond. Often, there may be more than one option for the decay of a nuclide.

Nuclear reactions can also cause transmutations. A nuclear reaction takes place when a nucleus of an atom is struck by another nucleus or by a particle such as a neutron which then causes a change to the nuclide. We are probably most familiar with nuclear fission as a reaction, but there are other types that we will examine and learn in a later lesson.

 

For Practice Problems, Try: Giancoli Multiple Choice Practice Questions (Don't worry if you can't solve all of them just yet!)