Binding energy per nucleon for atoms of low mass number

[kihr]

Could someone please explain to me as to why the binding energy per nucleon is lower for atoms of low mass number as compared to the atoms in the middle zone of the binding energy per nucleon versus mass number graph? My conceptual issue lies in the fact that when the mass number is low the coulomb force of repulsion between protons is also low and the internucleon attractive nuclear force should predominate over the coulomb force in these atoms. Hence the positive potential energy acquired when the nucleons are brought together is less than the negative potential energy due to the attractive nuclear forces (which is approx. constant in any atom because nuclear forces are short ranged). This difference between the two should be more for lower mass number. This should result in an even higher binding energy per nucleon than in the atoms in the middle zone of the graph. Kindly clarify / elaborate. Thanks.

 

[Useful nucleus]

One explanation for your question can be gained from the liquid drop model of the nucleus.This model predicts that the binding energy per nucleuon shuold be lower for lighter nuclei because of the surface effect. A surface nucleon would have lower binding energy compared to a "bulk" nucleon. The fraction of surface nucleons in light nuclei is higher , hence the lower B.E/nucleon.

On the other extreme of the curve which is the heavy nuclides, the columbic repulsion becomes very strong and outwieghts the fact that surface nucleons fraction is less in these heavy nuclides.

Intermediate nuclei has a balance between surface nucleons fraction and coulombic replusion , thus they tend to have the highest B.E/nucleon.

Refer to the semi-emperical mass fomula to get more quantitative insight about these factors.

 

[bcrowell]

This should probably be moved to the "High Energy, Nuclear, Particle Physics" forum.

Amplifying on what Useful nucleus said, the reason nucleons at the surface are less bound is that the strong nuclear force is attractive and short range.

 

[kihr]

Could you please quote the semi-empirical mass formula or refer me to the relevant web site? Thanks.

Regarding the nuclear force it should be the same irrespective of the mass number given its very limited range (up to about 2.5 to 3 femtometres). In a heavy nucleus the coulomb repulsion force should be higher compared to nuclei in the middle range because of the larger number of protons. The net attractive force would thus be lower in this case than for nuclei in the middle range.
However, I am not sure as to whether the same logic could be extrapolated to explain the lower BE per nucleon for the lighter nuclei compared to nuclei in the middle range.

 

[the_house]

The http://en.wikipedia.org/wiki/Semi-empirical_mass_formula" article doesn't look to bad. It explains each term in the semi-empirical mass formula.

The Coulomb term is as you describe in your first post, while what you describe in your last post corresponds to the volume term. However, you're ignoring the surface term. The nucleons near the surface don't have as much binding energy associated with them as a nucleon in the middle of a nucleus that is completely surrounded by other nucleons. Smaller nuclei have a larger ratio of surface to volume, and so they tend to have a smaller binding energy per nucleon.

 

[kihr]

Thanks a lot for providing the clues.