2.1.3 Nuclear Structure-Binding Energy
Created October 6, 1995
2.1.3.1 Binding Energy
When we listen to TV sports commentators discussing teamwork, they sometimes say that the team's performance is more than the sum of the parts, implying some synergistic effect of their "chemistry." In the nucleus, we have just the opposite going on - the better "team" that the nucleus is, the more tightly bound and energetically stable, the less mass it has. This is because of the mass/energy equivalence of Einstein, E=mc**2. A good nuclear "team", i.e. a well-bound nucleus such as He-4, is a whole lot less than the sum of its parts!
2.1.3.2 Total Binding Energy of a Nucleus
If we have a nuclear of mass m(A,Z), then we can write the conservation of mass/energy law for the reaction in which we break up this nucleus into its constituent nucleons as:
m(A,Z) + BE/c**2 = (A-Z) m(n) + Z m(p)
where m(n) and m(p) are the masses of the neutron and the proton, respectively. The term BE/c**2 is the total binding energy of the nucleus, expressed in units of mass. When the total binding energy is expressed in terms of mass such as this, it is known as the Mass Defect or Mass Deficit.
2.1.3.3 Average Binding Energy per Nucleon
The total binding energy of the nucleus goes up as mass number goes up. In order to compare the strength of each nucleus' binding, we must divide out the number of nucleons. In doing so, we find the AVERAGE BINDING ENERGY PER NUCLEON, which measures a nucleus' stability.
The plot of binding energy per nucleon is perhaps the most important physics graph we have discovered. This is the plot:
This plot is very important because it shows that 1) as atomic mass number is reduced from the highest values the binding energy per nucleon increases, which accounts for nuclear fission or the splitting of heavy nuclei; and 2) as atomic mass number is increased, the binding energy per nucleon also increases which accounts for nuclear fusion, the process responsible for the formation of the earth's elements from the primal sun's hydrogen and helium.
Note that the helium-4 (He-4, Z=2) nucleus has particularly high binding energy per nucleon - it is particularly stable. This particular nuclide is known as the alpha particle and is involved in radioactive decay of heavy nuclides. The alpha particle has a very low energy state due to the pairing of nucleons and the low number of nucleons involved.
2.1.3.4 Mass Excess- Capital Delta
Since the atomic mass unit mu is defined in terms of the 1/12 of the mass of the C-12 atom, it is useful to compare a particular nucleus' binding energy to that of C-12. The expressing for capital delta D is simply (M-A), where M is the mass of the nucleus in amu in question and A is the number of nucleons comprising the nucleus.
For C-12, M is defined to be 12 by the definition of amu, and A=12. D is zero in this case. For very tightly bound nuclei, D is negative (the mass is "lighter" than C-12). For weakly bound nuclei, D is positive (the mass is "heavier" per nucleon than in C-12).
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Douglas J. Wagenaar, Ph.D., wagenaar@nucmed.bih.harvard.edu