Energy and Decay Modes

The full chart of all the isotopes is called the Nuclide Chart. Watch the video below for an introduction to the Nuclide Chart, presented by Dr Christian Diget from the University of York. 

In the following video, Hannah Willett, a researcher at the University of York, introduces the key concepts about fusion in stars and on Earth. You will also be introduced to how fusion energy can be calculated based on the nuclear energy available in individual nuclear isotopes.

The hydrogen to neon nuclide chart shows the mass excess per nucleon, (in joules per kilogram) for all known isotopes from hydrogen to neon. The number in each square on the linked chart indicates the tower height (the number of layers of LEGO bricks). Each layer of LEGO bricks corresponds to 25 TJ/kg (1 TJ = 1x1012 J). Using Einstein’s equation, we can alternatively see this as the (mass) excess energy per kg of the material in question. We can explore this energy for all known isotopes on the Nuclide Chart.

The mass excess is the difference between the atomic mass of the isotope and the mass of the nucleons, (protons and neutrons) that makes it up:

mass excess = matomic(A, Z) − A x u

i.e. A = Z + N is the number of nucleons. All terms are in units of the atomic mass unit: the atomic mass unit (u) is one twelfth of the mass of an atom of carbon-12, (in kg, u = 1.66 x 10-27 kg). So, we are in fact comparing atomic masses with carbon-12 as a reference.

For example, deuterium (2H) has an atomic mass of 2.014u. Deuterium is composed of one proton and one neutron, so the mass excess of deuterium is 0.014u. We can also calculate the mass excess per nucleon (i.e. 0.014u / 2 = 0.007u for deuterium). To calculate the energy available from this mass excess, we use Einstein’s Energy Equation:

E = mc2

where E is the energy available, m is the mass excess, and c is the speed of light. So, E = 1.05 x 10-12 J.

In our LEGO nuclide chart, we use the mass excess for the tower heights because the most stable isotope is the isotope with the lowest mass excess. Therefore, on our chart, the most stable isotope is at the bottom of the 'valley of stability', and beta decay (plus and minus) will go towards the bottom of this 'valley'. However, we may also want to calculate the binding energy or nuclear mass defect of the isotope. These do not compare the mass of the isotope to the mass of the carbon-12 atom. In contrast, the mass defect and binding energy are defined by comparing the mass of the nucleus to the mass of the protons (mp) and neutrons (mn) in the nucleus

For a nucleus with Z protons and N neutrons, and a nuclear mass of M(Z,N), the nuclear mass defect is:

nuclear mass defect = (Zmp + Nmn) - mnucleus

The binding energy for the same nucleus is defined as:

binding energy = (nuclear mass defect) x c2 

binding energy = ((Zmp + Nmn) - mnucleus) x c2

Since almost all nuclei we would deal with are bound, the binding energy of these nuclei must be positive. The nuclear mass defect is therefore also positive in almost all cases. Over future modules, we will explore the binding energy much further, and even use it to predict the minimum size of a neutron star in Module 3 (Nuclear Astrophysics).

Radioactive decay is a random process.  This means that we cannot predict how long it will take an individual nucleus to decay.  Instead, we talk about the half-life of an isotope. This is the average time taken for half of the radioactive nuclei to decay (or the time taken for the count-rate to half).  

This carbon-14 reacts with oxygen to create radioactive carbon dioxide. Plants naturally absorb the radioactive carbon-14 from this carbon dioxide during photosynthesis. Animals eat the plants and therefore also consume the carbon-14.

The ratio of carbon-14 to carbon-12 is almost constant in all living things. Carbon-14 decays through beta-minus decay:

(where  ̅νe is an electron antineutrino) but is replaced at a constant rate, as long as the plant or animal is alive. But, when a living organism dies, it stops taking in carbon-14.  Therefore, the carbon-14 decays and is not replaced. This means that the ratio of carbon-14 and carbon-12 is no longer constant. We can use this to date objects made of materials that were once alive. 

This is explained further in the video below by archaeologist Dr Penny Bickle from the University of York: