Magic Numbers

Many stable atoms have ‘magic numbers’ of protons and neutrons − 75 years ago, two physicists discovered their special properties

Published: October 7, 2024 7.55pm BST

This is the article that alerted me to the existence of these so-called "magic numbers".

The word magic is not often used in the context of science. But in the early 1930s, scientists discovered that some atomic nuclei – the center part of atoms, which make up all matter – were more stable than others. These nuclei had specific numbers of protons or neutrons, or magic numbers, as physicist Eugene Wigner called them.

The race to figure out what made these nuclei so stable began. Understanding these magic numbers would allow scientists to predict the properties of other nuclei, such as their mass or how long they are expected to live. With that, scientists could also predict which combinations of protons and neutrons can result in a nucleus.

The solution to the puzzle came in 1949 from two directions simultaneously. In the U.S., physicist Maria Goeppert Mayer published an explanation, at the same time as a group of scientists led by J. Hans D. Jensen in Germany found the same solution.

For their discovery, the two physicists each got a quarter of the 1963 Nobel Prize in physics. We’re two nuclear scientists whose work is built on Goeppert Mayer’s and Jensen’s discoveries 75 years ago. These magic numbers continue to play an important role in our research, only now we can study them in nuclei that live for just a fraction of a second.

Stability in the atom

The atom is a complex system of particles. It’s made up of a central nucleus consisting of protons and neutrons, called nucleons, with electrons orbiting around the nucleus.

Nobel prize-winning physicist Niels Bohr described these electrons in the atom as existing in a shell structure. The electrons circulate around the nucleus in particular energy levels, or orbits. These orbits have specific energies, and each orbit can hold only so many electrons.

Chemical reactions result from interactions between the electrons in two atoms. In Bohr’s model, if an electron orbit is not already filled, then it’s easier for the atoms to exchange or share those electrons and induce chemical reactions.

The Bohr model of the atom.

One class of elements, the noble gases, hardly ever react with other elements. In noble gases, the electrons occupy completely filled orbits, and as a result the atoms greedily hold onto their electrons instead of sharing and undergoing a chemical reaction.

In the 1930s, scientists wondered whether protons and neutrons might also occupy orbits, like electrons. But nobody could show this conclusively. For more than a decade, the scientific community was unable to describe the nucleus in terms of individual protons and neutrons. Scientists used a more simplified picture, one that treated protons and neutrons as one single system, like a drop of water.

In 1949, Goeppert Mayer and Jensen developed the so-called shell model of the nucleus. Protons and neutrons occupy particular orbits, analogous to electrons, but they also have a property called spin – similar to a spinning top. Goeppert Mayer and Jensen found that when combining the two properties in their calculations, they were able to reproduce the experimental observations.

Through some experiments, they found that nuclei with certain magic numbers of neutrons or protons are unusually stable and hold onto their nucleons more than researchers previously expected, just like how noble gases hold onto their electrons.

The magic numbers known to scientists are 2, 8, 20, 28, 50, 82 and 126. They are the same for both protons and neutrons. When a nucleus has a magic number of protons or neutrons, then the particular orbit is filled, and the nucleus is not very reactive, similar to the noble gases.

For example, the element tin has a magic number of protons. Tin always has 50 protons, and its most common isotope has 70 neutrons. Isotopes are atoms of the same element that have a different number of neutrons.

There are nine other stable isotopes of tin that can exist – it’s the element with the largest number of stable isotopes. A stable isotope will never spontaneously change into a different element, which is what happens to radioactive isotopes.

Helium, with two protons and two neutrons, is the lightest “doubly magic” nucleus. Both its neutron count and its proton count are a magic number. The forces that hold the helium-4 nucleus together are so strong that it’s impossible to attach another proton or neutron. If you tried to add another proton or neutron, the resulting atom would fall apart instantaneously.

On the other hand, the heaviest stable nucleus in existence, lead-208, is also a doubly magic nucleus. It has magic numbers of 82 protons and 126 neutrons.

Many stable isotopes have magic numbers of protons and neutrons.

Examples of magic numbers and stable nuclei exist everywhere – but scientists couldn’t explain them without the introduction of the shell model.

Stable nuclei in nature

The shell structure in nuclei tells researchers about how elements are distributed across the Earth and throughout the universe.

One of the most abundant elements on our planet and in the human body is oxygen, in particular the isotope oxygen-16.

With eight protons and eight neutrons, oxygen-16 has an extremely stable nucleus. A nearby star produced the oxygen we find on Earth through nuclear reactions in its core sometime before the solar system was formed.

Since oxygen nuclei are doubly magic, these nuclei in the star did not interact very much with other nuclei. So more oxygen was left around to eventually act as an essential ingredient for life on Earth.

In her Nobel lecture, Maria Goeppert Mayer talked about the work she did with physicist Edward Teller. The two had attempted to describe how these elements formed in stars. In the 1930s, it was impossible for them to explain why certain elements and isotopes were more abundant in stars than others. She later found that the increased abundances corresponded to nuclei with something in common: They all had magic numbers of neutrons.

With the shell model and the explanation of magic numbers, the production of elements in stars was possible and was published in 1957.

Scientists today continue to use ideas from the nuclear shell model to explain new phenomena in nuclear science. A few accelerator facilities, such as the Facility for Rare Isotope Beams, where we work, aim to create more exotic nuclei to understand how their properties change compared with their stable counterparts.

At the Facility for Rare Isotope Beams, scientists produce new isotopes by accelerating stable isotopes to about half the speed of light and smashing them at a target. Out of the pieces, we select the rarest ones and study their properties.

Possibly the most profound modern discovery is the fact that the magic numbers change in exotic nuclei like the type we create here. So, 75 years after the original discovery, the race to discover the next magic number is still on.

If the sequence of numbers  2, 8, 20, 28, 50, 82,126 is entered into the OEIS, we find A018226 :

A018226     Magic numbers of nucleons: nuclei with one of these numbers of either protons or neutrons are more stable against nuclear decay.

The OEIS comments state:

"The results of the experiment indicate that 54Ca's first excited state lies at a relatively high energy, which is characteristic of a large nuclear shell gap, thus indicating that N = 34 in 54Ca is a new magic number, as predicted theoretically by the University of Tokyo group in 2001. By conducting a more detailed comparison to nuclear theory the researchers were able to show that the N = 34 magic number is equally as significant as some other nuclear shell gaps." Link

So it seems that maybe 34 should be included as well. It's interesting that the discovery was made in my birth year, 1949, now 75 years ago. So the sequence currently may better be represented as:$$2, 8, 20, 28, 34, 50, 82,126$$

Perimeter Magic Polygons

My previous post was titled Anti-Magic Squares Revisited in which I mentioned heterosquares. Only today however, I came across the concept of a perimeter magic square that is an example of a perimeter magic polygon or PMP to use an acronym. The definition is:

A PMP is defined to be a regular polygon with the consecutive positive integers from 1 to N placed along the perimeter in such a way that the sums of the integers on each side are constant. The order of a polygon refers to the number of integers along each side. The examples in Figures 1, 2 and 3 show a 4th-order triangle, and a 3rd-order square and pentagon. The magic constants are given inside the figures. Source.

Let's look at some perimeter magic triangles to begin with. To quote from Wikipedia:

A magic triangle or perimeter magic triangle is an arrangement of the integers from 1 to \(n\) on the sides of a triangle with the same number of integers on each side, called the order of the triangle, so that the sum of integers on each side is a constant, the magic sum of the triangle. Unlike magic squares, there are different magic sums for magic triangles of the same order. Any magic triangle has a complementary triangle obtained by replacing each integer \(x\) in the triangle with \(1 + n − x\). See Figure 4.

Figure 4

The author of this paper comes up to two sets of formula in which \(C\) is the magic constant,  \(n\) is the order of the polygon and \(k\) is the number of sides. Figures 5 and 6 show these.

Figure 5 shows the formulae for the case of \(n\) even or both \(n\) and \(k\) odd.

Figure 5

Figure 6 shows the formulae for the case of \(n\) is odd and \(k\) is even.

Figure 6

To quote from the article:

With these formulas one can now begin to construct PMPs of any order and number of sides. One word of caution is still in order. There are times when a solution is not possible for certain values of C. The formulas only serve to indicate where solutions may be found, and that there is no need to look elsewhere. However, luck is still with the solver. Based on this author's experience, there are only two cases where solutions cannot be made with values obtained from the formulas. These are 4th-order triangles with \(C \) = 18 and 22, and 3rd-order pentagons with \( C \) = 15 and 18.

By controlling certain of the variables, one can discover some interesting patterns that permit the rapid construction of special PMPs. For example, for 3rd-order PMPs with an odd number of sides, it can be proved that a solution s always possible for the minimum \(C \). And it can be further demonstrated (to the amazement of your friends) that you can produce the solution just as rapidly as it takes to write the \(N\) integers. Rather than present the proof here, two examples will be given, and you can easily note the pattern for yourself.

Figure 7 shows the two examples. 14 is the minimum value of \(C\) for \(n\) = 3 and \(k\) = 5. 19 is the minimum value for \(n\) = 3 and \(k\) = 7.

Figure 7

The pattern in easily perceived once you look at the numbers closely. The article goes further into how to create various PMPs and reference should be made to that for further examples. 

I stumbled upon this topic by way of the number associated with my diurnal age today, 27417. This number is a member of OEIS A135503 for the case where \(n\) = 38.

A135503

\( \text{a} (n) = \dfrac{n \cdot (n^2 - 1)}{2} \)

The initial members of the sequence are:

0, 0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

The OEIS comments state that for \(n\) > 2, a(\(n\)) is the maximum value of the magic constant in a perimeter-magic \(n\)-gon of order \(n \). For the case of \(n\) = 38, \(n\) is even and so the formula in Figure 5 applies. Substituting in \(n\) = 38 and \(k\) = 38 does indeed give the maximum value of the magic constant as 27417.

Anti-Magic Squares Revisited

I've written about magic squares before. Here are links to these posts:

• Ultramagic Squares on November 12th 2021
• Prime Semi-Magic Squares on May 17th 2021
• Magic Squares on September 21st 2020
• Anti-Magic Squares on July 17th 2018

In this post, I intend to revisit anti-magic squares and the numbers associated with OEIS A117560:

A117560

\( \text{a}(n) = \dfrac {n \cdot (n^2-1)}{2} - 1 \)

The OEIS comments state that:

\( \text{a}(n-1) \) is an approximation for the lower bound of the "antimagic constant" of an antimagic square of order \(n\). The antimagic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. 

The initial members of this sequence are:

2, 11, 29, 59, 104, 167, 251, 359, 494, 659, 857, 1091, 1364, 1679, 2039, 2447, 2906, 3419, 3989, 4619, 5312, 6071, 6899, 7799, 8774, 9827, 10961, 12179, 13484, 14879, 16367, 17951, 19634, 21419, 23309, 25307, 27416, 29639, 31979, 34439, 37022

27416 is highlighted because it is my diurnal age today and when I made my post about anti-magic squares on July 17th 2018, I was 25307 days old. This number immediately precedes 27416, a gap of 2109 (about 5.77 years in terms of days counted). It will be 2223 days, about 6.09 years, before the next number is reached.

The current number, 27416, relates to an approximation of the lowest integer of a 38 x 38 anti-magic square. Note that it is not necessarily the lowest, it's just an approximation. My July 2018 entry is quite thorough and there's no point repeating all the content there but Figure 1 shows an example of 4 x 4 anti-magic square just to reinforce the property of such a square.

Figure 1: source

The ten sums from a sequence of consecutive numbers, namely 30, 31, 32, 33, 34, 35, 36, 37, 38, 39. Figure 2 shows a different arrangement. Note that OEIS A117560 gives 29 as the lower bound here.

Figure 2: source

Note that an anti-magic square differs from a so-called heterosquare. As explained in Wolfram Mathworld: 

A heterosquare is an \(n \times n\) array of the integers from \(1\) to \(n^2\) such that the rows, columns, and diagonals have different sums. By contrast, in a magic square, they have the same sum. There are no heterosquares of order two, but heterosquares of every odd order exist. They can be constructed by placing consecutive integers in a spiral pattern (Fults 1974, Madachy 1979). An antimagic square is a special case of a heterosquare for which the sums of rows, columns, and main diagonals form a sequence of consecutive integers.

Figure 3 shows an example of a 4 x 4 heterosquare:

Figure 3: source

These numbers do not form a sequence of consecutive integers and so they do not form an anti-magic square.

Polydivisible or Magic Numbers

Today I turned 26440 days old and discovered a new type of number, one termed polydivisible or magic. Here is a definition from Wikipedia:

In mathematics a polydivisible number (or magic number) is a number in a given number base with digits \(abcde \dots \) that has the following properties:

• Its first digit \(a\) is not 0.
• The number formed by its first two digits \(ab\) is a multiple of 2.
• The number formed by its first three digits \(abc\) is a multiple of 3.
• The number formed by its first four digits \(abcd\) is a multiple of 4.
• etc.

To put it in formal mathematical terms, we have:

Let \(n\) be a natural number, and let \(k = \lfloor \log_{b}{n} \rfloor + 1\) be the number of digits in \(n\) written in base \(b\). The number \(n\) is a polydivisible number if for all \(0 \leq i < k\):$$\frac{n - (n \bmod b^{k - i - 1})}{b^{k - i - 1}} \equiv 0 \pmod i$$Using the initiall less formal, definition it can be seen that 26440 qualifies because:

• the first digit 2 is not 0
• the number formed by its first two digits 26 is a multiple of 2
• the number formed by its first three digits 264 is a multiple of 3
• the number formed by its first four digits 2644 is a multiple of 4
• the number formed by its first five digits 26440 is a multiple of 5

The polydivisible numbers in base 10 form OEIS A144688 and though defined differently, it amounts to the same thing:

A144688

"Magic" numbers: all numbers from 0 to 9 are magic; a number >= 10 is magic if it is divisible by the number of its digits and the number obtained by deleting the final digit is also magic.

The OEIS comments tell us that there are exactly 20457 terms, the largest of which is the 25-digit number 3608528850368400786036725. For any given base \(b\), there are only a finite number of polydivisible numbers. After 26440, the next polydivisible number is 26445 followed by 26480, 26485 and then a relatively large gap to 26720. In the range up to 26440, about 2.7% of numbers are polydivisible.

Figure 1 shows the distribution of the number of polydivisible numbers with \(n\) digits. These numbers form OEIS A143671.

Figure 1: permalink

From Wikipedia we learn that:

Polydivisible numbers represent a generalisation of the following well-known problem in recreational mathematics:

Arrange the digits 1 to 9 in order so that the first two digits form a multiple of 2, the first three digits form a multiple of 3, the first four digits form a multiple of 4 etc. and finally the entire number is a multiple of 9.

The solution to the problem is a nine-digit polydivisible number with the additional condition that it contains the digits 1 to 9 exactly once each. There are 2,492 nine-digit polydivisible numbers, but the only one that satisfies the additional condition is:

381 654 729

Numberphile made a video about this polydivisible number.

Wikipedia also list the following problems associated with polydivisible numbers:

• Finding polydivisible numbers with additional restrictions on the digits - for example, the longest polydivisible number that only uses even digits is 48000688208466084040.
  
  
• Finding palindromic polydivisible numbers e.g. the longest palindromic polydivisible number is 30000600003.
  
  
• A common, trivial extension of the aforementioned example is to arrange the digits 0 to 9 to make a 10 digit number in the same way, the result is 3816547290. This is a pandigital polydivisible number that includes 0.