Magic Numbers in Physics


The discovery of these so-called “magic numbers” was made primarily by Maria Goeppert-Mayer (1906-1972). Maria was the only the second woman to ever win the Nobel Prize in Physics, a feat she achieved in 1963. She had set out in 1948 to ascertain why nuclei with certain numbers of neutrons and protons appeared to be more stable than nuclei with different numbers of the same elementary particles.


At the time, such study was still very much in its infancy.  Most scientists of the period believed that the nucleus behaved like the drops of a liquid.  This model was particularly helpful for explaining the different aspects of the fission process.  It also allowed what is called the “semiempirical mass formula,” which designates a relationship between the atomic number of an element and its atomic mass.  Another popular model was created by Fermi, which treats the various nucleons as if they were particles of gas.  The advantages of this theory included an ease of explaining the tendency of nuclei to have even numbers of protons and neutrons, and also used the idea of the tendency of nuclei to occupy the lowest energy levels.


Goeppert-Mayer’s theories, however, uprooted the predominant theories of the time.  She analyzed many different elements and discovered that these “magic numbers” of protons and neutrons demonstrated greater stability than  elements with other numbers.  This led to the formation of the “shell” theory of the nuclei, which she presented in her paper in 1950 and which is still is the preeminent theory today.  (Incidentally, this paper confused many Russian scientists, because the initial translation of her paper translated the word “shell” as “grenade!”)  While another German scientist, Hans Jensen, made a similar discovery concurrently, though without her assistance, and he did share the Nobel Prize with her for their discovery, history has granted  Goeppert-Mayer the distinction of being the theory’s originator.  The two of them coauthored and published in 1955 their book on this theory, entitled Elementary Theory of Nuclear Shell Structure.


Goeppert-Mayer, however, did not coin the term “magic numbers.”  A contemporary of hers, Eugene Wigner, applied this nickname to these numbers.  He felt that the theory behind the numbers was without concrete evidence and did not believe her findings.  Later, however, he too would concede the truth in Goeppert-Mayer’s model.


It is now agreed that the magic numbers for both protons and neutrons are 2, 8, 20, 28, 50, and 52. For protons, 114 is also a magic number; for neutrons, 126 and 184 are also magic numbers. A nucleus with magic numbers for both protons and neutrons is said to be “doubly magic.”  The reasons for these numbers are very complicated, and involve the paring of nucleons by the Pauli exclusion principle.  Once all possible sets of quantum number assignments have been filled up, then the instability is created. 


Research into magic numbers has extended into other areas. In 1998, two German and one Chilean physicist experimented with swirling spheres in a dish. They discovered that with fifty-four or fewer spheres, there were certain numbers which produced solidlike shell structures. These magic numbers are 7, 8, 12, 14, 19, 21, 30, 37, and 40. Other numbers of spheres did not exhibit the geometrical patterns of the magic numbers when swirled.  It is this experiment, which was originally done by computer simulation, that I will try to replicate in real life.




I wanted to discover how the aforementioned model of solidlike cell structures could be replicated cheaply and performed in a manner that the average Physics student could understand. 




Initially, I felt that if I were to use marbles to represent the spheres and then use a hollow-bottomless cylinder to act as the dish, one could demonstrate these properties by, say, using an overhead projector and placing the marbles upon the surface of the projector.  I theorized, additionally, that as the numbers of marbles got further from the magic numbers, the disorder would increase.



When I began, I first tried to generate this demonstration using a hollow cylinder with no bases as my container for the marbles.  On the overhead projector, therefore, I would be able to have a clear image with the walls being apparent.  This proved ineffective.


I am not certain as to why this was so ineffective, but it is my intuition that this is because I was attempting to illustrate this concept in two-space.  The problem with this is that when we do perform this operation on a single plane, there is still a loss in momentum for each of the spheres when they collide with the walls of the container. 


To compensate for this, I found bowls which were basically cross-sections of a sphere, with the base a plane.  This allowed the final product to still exist in one dimension, but the balls would be supplied with additional momentum from their rise and fall with the motion. 


The experiment I researched talked much of “swirling.” However, it was never very well defined what “swirling” really was. I presumed first that this swirling was intended to be in two dimensions, and thus a spiraling motion.  Therefore, I set up the following apparatus:


This succeeded in creating a semi-spiral motion that would work, but due to the inevitable effects of centrifugal force, the marbles were continually clustered to the edges.


At this point, I abandoned all attempts to create a perfect situation.  Realizing that the simulation the experiment was done in was highly idealized, I attempted to achieve similar results by any means necessary.  To my happy surprise, I was indeed able to do so.

I videotaped the evidence of my demonstration, since I was unable to mechanically create the circumstances necessary for such a demonstration.  In this demonstration, the following trends can be observed:


·                    As the number of balls approaches the magic number 7, we begin to see more and more semblances of order. When 7 is finally reached, we see a 4-3 pattern emerge.

·                    With 8, the pattern is not as visible, but cohesion of the elements is still noticeable.

·                    Once 9 is reached however, we start to see that at least one of the marbles does not move with the others.

·                    Arriving at 10 marbles, we notice that order has degenerated greatly.


These trends I observed confirm my hypothesis that disorder increases as distance from the magic number increases.




Performing this experiment was useful in that I was able to attempt to design an experiment an implement it.  Despite the fact that I may have failed this attempted mission twice by engineering ineffective models, I was still able to observe the properties of theoretical physics in a real life situation in the end.  While the confirmation of my hypothesis will require further experimentation before I can assert it as an absolute truth, I did at least confirm the observations in the simulated experiment I read about that the numbers 7 and 8 did produce more order than the others.  Even in light of my relative unsuccessfulness in designing an experiment, I very much enjoyed this more creative aspect of science, which I usually find so lacking in science classes in general (though AP Physics has presented an unusually high number of creative opportunities). I look forward to designing similar experiments in the future, perhaps with more sophisticated equipment to ensure a higher quality of results.




So what applications are there to the idea of magic numbers?  Though they were applied to Physics almost half a century ago, we are still trying to uncover their true meaning and their applications. 


One practical application I learned about was discovered by a team at Oak Ridge National Laboratory.  In May 1998, they began to use the principle o f magic numbers to grow thin metal films on semi-conductors.  Further research in this area could lead to the production of certain films that will be necessary for the development of future technology in the field of electronics.


In terms of a theoretical application, in September 1999 a team of scientists in France were able to discover nickel-48, which features a “doubly magic” nucleus.  This is probably the last of its kind yet to be observed.  The observations of the special properties of this nucleus sheds light on the basic mechanism involved in the cohesion of atomic nuclei and will undoubtedly lead to new discoveries and applications in the future.