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Binding Energy
Origin of Elements
The Liquid-Drop and The Shell Models
Nuclear Decay
Nuclear Fission
     Release of Binding Energy,
     Theory of Fission,
     Neutron Induced Fission,
     Cross Section,
     Chain Reaction and Critical Mass
Applications of Nuclear Fission
     Nuclear Fuel Production,
     Nuclear Reactors,
     Nuclear Bombs
Thermo-nuclear Fusion
Applications of Thermo-nuclear Fusion
Effects of Nuclear Explosions
The Fukushima Incident

Binding Energy

Binding Energy Proton-Neutron Ratio A nucleus is specified by its number of protons Z, number of neutrons N, and the mass number A = Z+N. The nucleons (protons and neutrons) in a nucleus are bound together -- their total energy is less than the total energy of the separated particles. The binding energy is the amount of energy given up when the nucleus is formed. Plotting the binding energy per nucleon versus the mass number A (Figure 14-01) shows that starting from Hydrogen, nuclei become more stable as there are more

Figure 14-01 Nuclear Binding Energy
[view large image]

Figure 14-02 Proton/Neutron & Decay [view large image]

protons and neutrons, until Iron. After that, the trend reverses.

Figure 14-02 shows the distribution of the stable nuclei. As the mass numbers become higher, the ratio of neutrons to protons in the nucleus becomes larger. There are no stable nuclei with a mass number higher than 83 or a neutron number higher than 126. This limit is represented by the element Bismuth (see Figure 13-01b). Although it is not obvious from Figure 14-02 (due to its lack of detail) stability is favored by even numbers of protons and even numbers of neutrons. 168 of the stable nuclei are even-even while only 4 of the stable nuclei are odd-odd. Notice how the stability band pulls away from the P=N line. Figure 14-02 also shows all the trends of decay. There are some exceptions to the trends but generally a nucleus will decay following the trends (in multiple steps) until it becomes stable. This process is called a radioactive series. For example, the series for 92U238 will go through 8 alpha emissions and 6 beta emissions before becoming the stable nucleus 82Pb206.

    The curve of stable nuclei portrays in Figure 14-02 is the result of the balancing act between the various repulsive and attractive effects:

  1. Electric force (repulsive) : There is the obvious electric repulsion between the protons each carrying a positive charge.
  2. Uncertainty principle (repulsive) : According to this principle at short distance (equivalent to reduced uncertainty in position) the uncertainty in momentum becomes correspondingly large giving rise to higher kinetic energy and a tendency to disperse.
  3. Exclusion principle (repulsive) : Since identical particles cannot be in same state, close proximity between similar particles also implies higher kinetic energy and a tendency to disperse.
  4. Strong Interaction (attractive) : This is the short range nuclear force that operates on all the protons and neutrons. It is this force that holds the nucleus together.
  5. Neutrons (attractive or repulsive) : Adding neutrons to the nucleus tends to minimize the repulsive effects. However, too many of them in there would favor the beta decay reaction, which turns neutron into proton and makes the nucleus unstable. Isotopes are elements with varying number of neutrons but same number of protons in the nucleus. If there are too many protons in the nucleus, there is no way to add neutrons to overcome the electric repulsion. This kind of elements would become unstable via alpha decay or other processes to reduce the number of positive charge.
  6. There are theoretical limits to the number of protons in the nucleus regardless of forces. According to the Bohr's model the velocity of the electron v = Zc, where the fine structure constant = 1/137. Thus, the number of protons Z in the nucleus cannot excess 137 or the velocity would be greater than the speed of light in violation of special relativity. Under the more sophisticated Dirac equation, the limit of Z is 173 imposed by the rising of binding energy up to pairs creation, which renders the atom unstable. The heaviest nucleus to have been identified experimentally has Z = 118.
See more in "Nuclear Binding Energy" from Hyper Physics.

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