ATOMIC NUCLEUS: How Nuclei Change as a Result of Radioactive Decay, Mass Defect and Binding Energy #2

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EXPLAINING THE BINDING ENERGY CURVE

The curve of binding energy per nucleon (BEPN) in the figure below can be divided into three main parts:

  1. The steep slope at the start.
  2. The flat part close to the iron nucleus
  3. A steady climb to the limit of natural elements


Binding energy curve.Fastfission,public domain

In more detail:

  • Here, as we go from hydrogen towards iron nuclei, BEPN increases as more nucleons are added and each contributes to the strong nuclear force, so binding the particles more strongly. Hydrogen with just one proton has a binding energy of zero; this is like a free mass far away from any other gravitational attracting body. Adding a neutron to make deuterium increases the binding energy as the strong nuclear force now comes into action. Binding energy per nucleon increases rapidly (negatively) as particles are added, and this means that energy is being released. As explained above, adding nucleons to a nucleus is not easy, but happens in the extreme conditions found in the very hot ultradense cores of stars. This nuclear fusion is, of course, the source of a star’s energy output which keeps it hot and maintains the reaction. We can imagine nucleons and nuclei ‘falling downhill’ and converting potential energy (binding energy) to kinetic energy (‘heat’) and radiation as they do so.
  • Adding even more nucleons increases the binding forces but also increases the electrical repulsion effect as protons are added. This means that there comes a stage when adding more nucleons starts to decrease the BEPN. The turning point is iron, which has the greatest BEPN of all nuclides. No nucleons or nuclei can ‘fall’ further than iron. Just as it needs an input of energy to break up an iron nucleus into smaller nuclei (going up the steep initial slope), it also needs an input of energy to create nuclei larger than iron (going up the shallower slope to the right).
  • The shallow rising slope is the effect of electrical repulsion (protons) beginning to have an increasing effect on the net force holding the nucleus together. So it is sometimes called the Coulomb slope. There is a tendency here for nuclei to fall down the slope towards iron by emitting a very stable unit – the helium nucleus, which has a high BEPN. It is emitted as the familiar alpha particle. Very large nuclei with a weak net binding force (low BEPN) can reach stability by simply falling apart. This is rare but can happen in large nuclei with an excess of neutrons (e.g. U-235). It can be stimulated by adding neutrons to the nucleus – which is what happens in nuclear fission – in bombs and nuclear reactors.

All the nuclides to the right of iron on the graph have been made when stars have collapsed and subsequently exploded: the huge amounts of energy released have caused Nuclei to fuse into heavy nuclei. All the nucleons to the left of iron been made as a consequence of the energy-producing nuclear fusion that keeps stars hot during their lifetime. 

FROM TWO TO THREE DIMENSIONS

The BEPN curve in the figure above is drawn as two-dimensional. In fact, each point on it (representing an element) has partners with the same proton number (Z)- the element’s isotopes. The slope looked at in three dimensions is like a valley with steep sides – like a canyon. At any point, the isotopes of an element lying on either side of the most stable isotopes have smaller (negative) values of BEPN. This means that they can lose potential (binding) energy by falling down to the stable isotope at the bottom of the canyon. The isotopes tend to be unstable because of a faulty proton-neutron balance. On one side the isotopes have too many neutrons, on the other too many protons, for stability. Like electrons in atoms, these nucleons can’t have the same energy level, so their nuclei are at higher energy states – higher up the canyon wall. They gain stability by converting neutrons into protons – or protons into neutrons – in other words by beta decay.

In ‘ordinary’ beta decay, a neutron changes to a proton and an electron is emitted (β-):

10n → 11p + 1-1e

In positron decay, a proton changes to a neutron and a positive electron, called a positron, is emitted (β+):

11p → 10n + 01e

Gamma emission

It is useful to think of the nucleus as a kind of onion having layers of nucleons – rather like a simple model of an atom with its ‘layers’ of electrons. Just as atoms can be in excited states because electrons are at higher levels, so neutrons and protons in a nucleus can be left at higher levels when nuclei change quite dramatically by alpha decay or by nucleons changing via positive and negative beta decay. Atoms achieve stability as electrons move, emitting electromagnetic radiation as they do so. Nuclei do the same, but because the nuclear force is so strong the radiation emitted is highly energetic – the gamma ray.


Radioactive decay scheme of 60Co. Inductiveload, public domain

A REGION OF STABILITY

Along a narrow band at the bottom of the ‘canyon’ of the three-dimensional binding energy curve are nuclei with a balance of protons and neutrons that keep them stable. In general,as Z increases more neutrons than you might expect are needed to do this: the neutron-proton ratio increases for stable nuclei, from 1 for He to 1.6 for uranium. It is these stable nuclei that make up the ordinary chemical world. It is only the very long-lived unstable nuclei (like uranium-238, thorium-232) that still exist on Earth – unless they are fairly new nuclei formed by radioactive decay or artificially in nuclear reactors.

In a plot of neutron number against proton number which clearly shows that neutron number increases more rapidly than proton number for stable isotopes. If isotopes stray from this equilibrium line they tend to decay via beta+ or beta- routes r, more rarely, by a nucleus capturing an electron from the nearest electron orbital shell – electron capture.


Graph of isotopes by type of nuclear decay. Orange and blue nuclides are unstable, with the black squares between these regions representing stable nuclides. The unbroken line passing below many of the nuclides represents the theoretical position on the graph of nuclides for which proton number is the same as neutron number. The graph shows that elements with more than 20 protons must have more neutrons than protons, in order to be stable. Napy1kenobi, Cc-by-sa-3.0








WHY DON’T ALL UNSTABLE NUCLEI DECAY IMMEDIATELY?

Some radioactive nuclei decay so slowly that it takes billions of years for half of them to decay (e.g. U-238 with a half-life of 45 billion years). Others decay in fractions of a microsecond. What causes even the shortest decay? The answer is to do with the facts of quantum behaviour. As an example, consider alpha decay,

An alpha particle is so stable that it behaves as a single object inside a large nucleus. We can think of this particle in a nucleus as being at the bottom of a potential energy well. This is similar to picturing a mass on the Earth’s surface at the bottom of the Earth’s ‘gravitational well’. In both cases, there has to be some energy input for the object to reach an ‘escape speed’ and get away from the well.

But how does the alpha particle ‘escape’ from the well? The diagram below suggests that it needs to be given kinetic energy to get away from the other particles in the nucleus. It does have some kinetic energy – the nuclear particles are imagined as joggling about inside the nucleus like gerbils in a bag – but not enough to allow a particle to escape. The diagram shows a typical alpha particle with an energy of 5 MeV inside a nucleus. But the nuclear force is so strong that an energy of some 30 MeV is needed for the particle to escape. You can compare this with the idea of escape speed for a rocket leaving the Earth.

This is where the quantum theory of matter comes in. This says that, like a gerbil that eats through the side of the bag, an alpha particle could ‘tunnel’ its way out. In this theory, the position of an alpha particle is represented by a probability wave of a size that fits into the nucleus. Compare this with an electron inside an atom. But, as the probability wave doesn’t end inside the nuclear potential energy barrier; there is a small but finite probability that the alpha particle is outside the nucleus, where the electric repulsion force alone acts and returns the alpha particle’s kinetic energy as it accelerates away. The probability of decay – essentially the decay constant k which determines the half life – is different for different nuclei and as it is a probability the resulting decay is random.

Similar quantum behaviour decides the decay probabilities for beta and gamma emission.


A generic potential energy well. Benjamin D. Esham (bdesham), Public Domain

A RADIOACTIVE PUZZLE – AND ITS SURPRISING SOLUTION

Energy conservation and beta decay – the neutrino: In radioactive decay, both alpha particles and gamma radiation are emitted from a given nuclide with a definite energy which is characteristic of the nuclide. This, however, does not happen for beta particles. Measurements show a range of energies for the beta particles emitted from a given radioactive substance. 

This spread of energies posed a serious problem to nuclear physicists. Why did some emerge with less energy than others? It seemed to contradict the principle of the conservation of energy. To avoid this appalling prospect, Wolfgang Pauli suggested in 1930 that the missing energy was being carried away by a new subatomic particle. The particle had to have a very small mass, and, because no one had yet detected such an energetic particle, it must carry no charge. This particle has been named the ‘little neutron’, or neutrino (symbol v, the Greek letter ‘nu’). We also know that there is more than one kind of neutrino, each with a mass very much smaller than the mass of an electron – possibly zero. Reactions involving beta decay should include the neutrinos, e.g.

146C→ 147N + -1β- + 0

158O→ 157N + -1β+ + 0v

In these reactions,v is the ‘ordinary’ neutrino, and ṽ is its antiparticle, called the antineutrino. When a neutron decays to form a proton and a negative beta particle, the antineutrino is involved:

1n → 1p + β- + ṽ

Note that the beta decays earlier shown under the positron decay are incomplete: neutrinos are also emitted.

In the C-14 decay above, a neutron has changed to a proton; in the O-15 decay, a proton has changed to a neutron. Deeper inside the nucleus quarks are changing to cause these effects.

In summary, there are six main ways in which unstable nuclei decay or change:

  • Alpha decay (mostly very large nuclei; A and Z change), e.g. 

23290Th → 22888Ra + 42α

  • Negative beta decay (nuclei that have too many neutrons; Z changes), e.g. 

146C → 147N + -1β- + ṽ

  • Positive beta decay (nuclei with too many protons; Z changes), e.g. 

106C → 105B + 1β+ + v

  • Electron capture by nucleus of an electron in innermost shell (some nuclei that have too many protons; Z changes), e.g. 

5425Mn + -1e-5424Cr + γ

  • Gamma emission (nuclei left in an excited state after decay; no change in A or Z but a change in energy level)
  • Nuclear fission (some very large nuclei, particularly after neutron capture), e.g. 

235U → 100Ru + 133Cs + 2 1n


REFERENCES

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