Week of November 1, 2021

Missing in the historical story of nuclear theory is a straightforward connection between the geometric arrangement of particles and the instability of nuclei. Though some anecdotal proposals of connections between such geometric arrangements of protons, for example, an real nuclear phenomena have cropped up in the last century, they all fail to be anything more than wishful thinking. Here, geometric distributions of discrete point charges in the Thomson problem -- the lowest-energy configuration of electrons on the surface of a sphere as they interact through Coulomb's force law -- are shown to exhibit features that one may argue is a fingerprint of atomic structure. If we consider energy difference among solutions of the Thomson problem, we find correlations with the first ionization energies of atoms (a subject to be taken up in greater detail in a later blog entry). If we consider this energy difference a discrete first derivative of the energy, then this blog entry is all about the second discrete derivative of the energy of the Thomson problem. More practically, we already have the nomenclature borrowed from Chemistry for this sort of thing -- the chemical hardness.

Chemical Hardness

In chemistry is the concept of chemical hardness originated by the chemist R. G. Pearson. Chemical hardness is a useful concept that allows chemists to understand reactivities commonly encountered in the hard-soft acid-base (HSAB) theory for Lewis acids and bases -- Chemistry 101. The premise is a comparison between the amount of energy needed to add a single charge to a system and the amount of energy needed to remove a single charge from a system. Originally, Pearson proposed the following expression for the chemical hardness,

in which I is the ionization energy (due to loss of a single charge) and A is the affinity energy (due to the gain of a single charge). Later Pearson noted that he merely wanted to "symmetrize" the expression with the expression for electronegativity, written

.

He further agreed that more recent and frequent use of the expression for chemical hardness,

by many chemists was appropriate and practical. (I'll add a reference to substantiate these remarks at a later date. But I recall these comments made perhaps in a lecture published in the late 1990s.)

Charge Hardness

Let us suppose we apply the concept of chemical hardness to the comparative process of adding or removing a single electron from the Thomson problem. We might rename this particular hardness quantity an "electron hardness," but let us be more general about things and instead label it simply as the "charge hardness". Thus, we can evaluate the charge hardness by subtracting the energy needed to add a charge to the Thomson problem from the energy needed to remove a charge from the Thomson problem. If we then plot the charge hardness for perhaps the first N=100 charge solutions of the Thomson problem, we find the distribution shown in the graph below.

Though the plot of data in the graph appears to be a random distribution of points, this is very much a systematic distribution -- one that will be discussed in a later blog entry. In fact, the error bars associated with each plotted data point is much smaller than the symbol displayed. This is because the numerical energy solutions of the Thomson problem are precise to as many as 6 to 9 decimal places for up to N=100 charges.

A Personal Note

When I first examined this set of data, I was puzzled by the appearance of the tight string of data points between N=83 and N=87. I was initially preoccupied with attempting to understand the data in the context of electrons in atoms (and there are some interesting electron structure-related observations I'll share in a later blog entry). The following day I happened upon a figure caption in my undergraduate physics textbook by Halliday and Resnick[1] that noted that bismuth (atomic number 83) is the largest known stable element. This coincides very well with the distribution of charge hardness values in the Thomson problem. In other words, the amount of energy needed to add or remove a proton to the N=84 Thomson problem solution is negligibly different. So, the proton experiences no energetic preference to reside in the system versus not residing in the system. The system is therefore susceptible to falling apart just as unstable elements lose a proton or two during radioactive beta and alpha decay processes, respectively.

There is a refinement to this discussion that will need to wait for a later blog entry for explanation, but think about the physical meaning of a line that would pass through the vertical average of the distribution of points in the plot above and note that the string of charge hardness values in the N = 83 - 87 region are very near this average. It has much to do with geometry -- or better, quality of the geometry (geometric symmetry).

To some varying degree, it may be possible to "map" known radioactive decay chains of certain isotopes through the plot of charge hardness shown here if we consider the charge hardness in terms of the statistical probability of either alpha or beta decay, for example. For even protons in a nucleus, with adjacent neutrons, there appears to be some geometry-limited mechanism related to their radioactive decay and stability.

There are more nuclear phenomena with which the charge hardness of the Thomson problem appears to coincide. This includes the ratio of protons to neutrons as well as the so-called island of stability for larger atoms. Let's save those for a later blog entry.

Other Applications of Charge Hardness

The charge hardness in the Thomson problem exhibits some enticing features related to the electron structure of atoms as well. These phenomena include known exceptions to the rules of quantum mechanics and other electron shell-filling rules.

The charge hardness can also be found in a more general electrostatics framework, and understanding it helps us to understand the concept of so-called quantum capacitance related to the addition and removal of a single electron from a nanoelectronic device like a quantum dot. We'll take up this subject in a later blog entry as well. 

References

[1] Halliday & Resnick, Fundamentals of Physics, 3rd ed, Caption of Segre Chart, fig. 4, Section 47-2 "The figure shows that there are no stable nuclides with Z > 83 (bismuth)."