This week’s content is focused around the Nuclide Chart. This is the chart of all the nuclear isotopes.
1.1 Nuclear Decay and Super Heavy Elements (Questions)
1.2 Energy from Fusion (Questions)
1.3 Energy Calculations (Questions)
1.4 Modelling Radioactive Decay (using Excel or Google Sheets)
1.5 Activity, Decay and Half Lives (Questions)
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.
The colours in the chart shown in the video above represent different nuclear decay modes: beta decay (beta-minus and beta-plus), alpha decay, (a type of nuclear fission) as well as the more exotic proton emission and neutron emission.
The nucleus emits an alpha particle (a helium nucleus, made of two protons and two neutrons).
The mass number decreases by 4; the atomic number decreases by 2.
A neutron inside the nucleus turns into a proton, emitting a negatively charged electron, (hence beta-minus) and a very light particle called an anti-neutrino in the process.
The mass number does not decrease; the atomic number increases by 1.
A proton inside the nucleus turns into a neutron, emitting a positron, (the positively charged antiparticle of the electron) and a very light particle called a neutrino in the process.
The mass number does not decrease; the atomic number decreases by 1.
The nucleus emits a proton.
The mass number decreases by 1; the atomic number decreases by 1.
The nucleus emits a neutron.
The mass number decreases by 1; the atomic number does not decrease.
An unstable heavy nucleus splits to form two lighter nuclei, emitting neutrons as part of the process.
The mass number and the atomic number decrease.
The nucleus loses excess energy by emitting a photon.
The mass number and the atomic number do not decrease.
We have so far discovered over 3,000 isotopes! But, is the nuclear chart complete? Find out about potential super heavy elements with Dr Ulrika Forsberg from the University of York, in the video below.
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Super Heavy Elements
Nuclear Equations 3 to 10
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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.
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The Nuclide Chart and Energy From Fusion
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The height of each tower shown in the chart represents the mass excess of the isotopes relative to iron-56. Christian Diget explains this further in the video below.
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).
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Energy Calculations
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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).
If the initial number of radioactive atoms in a sample is N0, it is related to the current number of radioactive atoms, N, at time t by the equation:
where λ is the decay constant. At the half life, where t = T1/2, N = N0/2. So:
Cancelling out N0 from both sides gives:
Taking the natural log of both sides then gives you:
Rearranging this shows that the decay constant is related to the half life by:
The activity, A, of a sample is the number of decays per second, (measured in becquerel, Bq). It is related to the decay constant, λ, and the number of undecayed nuclei, N, by the equation:
Figure 1: A graph showing the decay rate of a hypothetical radioactive isotope. This substance has a half life T1/2 of 3.5 days, so, every 3.5 days, the number of radioactive atoms halves.
We can model radioactive decay using other random processes, such as tossing coins or rolling dice. Have a go at modelling radioactive decay using the random number generator in a spreadsheet. Instructions are available for Excel or Google Sheets:
One application of half lives is radiocarbon dating. This is the process of determining the age of a previously living object by measuring the ratio of carbon-14 to carbon-12 present in a sample, (carbon-14 is unstable, and decays to nitrogen-14, whereas carbon-12 is stable).
Carbon-14 is formed in the upper atmosphere. Cosmic rays enter the Earth’s atmosphere and collide with atoms. This creates an energetic neutron that collides with nitrogen-14 to produce carbon-14 and a hydrogen nucleus (a proton):
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:
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Modelling Radioactive Decay
Activity and Decay 1
Nuclear Decay with Time 1, 4, 5, 6, 8, 9
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The Colourful Nuclide Chart, created by Dr Ed Simpson from Australian National University, can tell you a lot about the properties of the over 3000 different isotopes that exist. To get the colours to match our LEGO(R) chart, click on the menu in the top left corner, select 'Colours' and then tick the box 'Force Binding Blocks decay mode colours'.
Try investigating the chart further. For example, in the menu, select 'Data' to choose whether to display the ‘Primary Decay Mode’, 'Mass Excess', or ‘Binding Energy per Nucleon’ (among other options). Under 'Display', you can also choose to view the chart in 3D.
We've built our Binding Blocks Nuclide Chart in Minecraft and made it available for download. To download the world, you will need either a Java edition or Bedrock edition of Minecraft. In the world you can explore the chart from Hydrogen, through the Valley of Stability, all the way to the super heavy fissile elements. Along the way, we've placed books for you to read about some of the more interesting isotopes and in our virtual Central Hall, you can also learn about the different types of decay. Click here to download the world.
If you're interested in a more in depth look at the nuclide chart, and nuclear physics in general, you may want to watch "The valley of stability" - a video produced by the CEA (French Alternative Energies and Atomic Energy Commission).
At the University of York, Dr. Marina Petri, a Royal Society University Research Fellow in Nuclear Physics, studies the structure of atomic nuclei using particle accelerators to probe nuclear properties with accelerated ion beams. This includes how the protons and neutrons arrange themselves within the nucleus and how they interact with each other to form complex nuclei and more energetic states of these nuclei, known as excited states. This research has shown incredible results about excited states of neutron rich carbon isotopes, such as the first excited state of carbon-20, whose lifetime was measured as only 0.000000000010 seconds (10 ps).
By contrast, the unexcited ground state of carbon-14 is extremely long-lived, with a half life of several thousands of years before decaying to nitrogen-14. The long lifetime of carbon-14 allows us to use radiocarbon dating to determine the age of historic artefacts and paintings. Based on the observed lifetimes of excited states in carbon-14, carbon-16, carbon-18, and carbon-20, Marina and her collaborators have furthermore shown how the six protons in the nucleus of carbon isotopes change their behaviour as more neutrons are added.
References: M. Petri et al. Lifetime Measurement of the 2+1 State in 20C, Phys. Rev. Lett. 107, 10250 (2011). Phenomenological analysis of B(E2) transition strengths in neutron-rich carbon isotopes, Phys. Rev. C. 90, 067305 (2014)