The curve of binding energy is a graph that relates nuclear binding energy to atomic mass number. The x axis variable is "Atomic Mass Number" and the y axis variable is "Binding Energy per Nucleon" (measured in MeV). It graphically describes why small nuclei release energy during fusion while large nuclei release energy during fission. This graph is sometimes shown inverted to better represent that iron is at the lowest energy state.

                CURVE OF BINDING ENERGY

         Exothermic       Endothermic
         Fusion           Fusion
        |---------|---------------------------|
              |--Stable--|
b    9 +
i p    |          *
n e    |      *          *
d r    |                        *
i    8 +  *                            *
n n    |                                      *
g e    | *
  u    |
e c  7 +*
n l    |
e e    |
r o    |
g n  6 +----+----+----+----+----+----+----+----+
y      0        50       100       150       200
(MeV)
                  atomic mass number

The graph begins at roughly (2 atomic mass, 2 MeV) (deuterium) and increases sharply up to about (25, 8) (magnesium), at which point it begins to level off until it peaks at roughly (60,9) (iron). It then follows a fairly linear down slope in the direction of about (200,8) (mercury) and continues along that slope. Note that this curve is an average trend, some isotopes do not fall perfectly on the line. The biggest discontinuity occurs at helium 4. The previous structure, tritium (hydrogen 3) has almost 3 MeV per nucleon, then helium 4 has over 7 MeV per nucleon, which drops to less than 5.5 MeV per nucleon at the next structure, lithium 6 (there are no stable nuclei with five nucleons). After this the binding energy follows the average trend much closer.

As described in the writeup on mass defect, as protons and neutrons come together to build a nucleus, they lose some of their mass when it is converted to energy and released. This energy is called binding energy, since this much energy must be reabsorbed by the nucleus in order to break it back apart. The higher the binding energy per nucleon, the more stable the nucleus is. Since iron 56 is the most stable nucleus, this is where the curve of binding energy peaks.

Up to iron 56, nuclear fusion is exothermic. This means that energy is released as the nucleus gets larger by adding more nucleons. Therefore stars can build these elements with their standard fusion process which provides light and heat to their solar systems. Since this means the binding energy per nucleon gets higher and higher, the nucleus becomes more and more stable.

Above iron 56, where the graph drops off, energy must be absorbed to create a larger nucleus because the binding energy per nucleon begins decreasing, resulting in decreasingly stable nuclei. Note that the total binding energy continues to increase, only the binding energy per nucleon begins to drop off. The star's normal fusion process cannot create these elements, in nature a supernovae can supply an enormous neutron flux to create stable elements up to uranium (238 atomic mass) through a process called endothermic fusion. Although it costs energy to build these large nuclei, energy is released when they are split. This is the principle behind nuclear fission reactions.

The one exothermic method by which transferric nuclei can be built is through slow neutron capture. Artificially, transuranic elements can also be created in a particle accelerator.

By the curve, then, we see that for small elements such as hydrogen, energy is released by fusion and energy is required to produce fission. Larger elements such as uranium require energy for fusion and release energy during fission. The large number of nucleons compared to the small binding energy per nucleon is what makes these larger elements radioactive — they can barely hold themselves together.

Sources:
http://csep10.phys.utk.edu/astr162/lect/energy/bindingE.html
http://www.tpub.com/content/doe/h1019v1/css/h1019v1_45.htm
http://www.colorado.edu/physics/2000/periodic_table/amu.html
http://www.cartage.org.lb/en/themes/Sciences/Chemistry/NuclearChemistry/NuclChemIndex/NuclearBindingEnergy/NuclearBindingEnergy.htm

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