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.
Problem
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.
Hypothesis
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.
Procedure
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.
Conclusion
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.
Applications
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