Having obtained a copy of my doctoral advisor's book, The World and I, and skimming through its photos I noticed that more than a decade after I graduated he was still using the language I invented for my doctoral work. In fact, he suggested that I had the opportunity to create some new language that was specific to the body of work I created. It was a way to compactly discuss the types of changes a system of electrons would undergo and it would thereby label the types of properties associated with these transitions. As I recall he set me off one day with this in mind after I quickly free-thought a few possibilities that I might want to take some time to think about it carefully and decide which type of language would fit best. So, I went off and came back the next day with some thoughts.

I had three different root words, if you will, that could potentially characterize the types of transitions and static properties involved in my work. The roots were

• phase
• mode
• state

Privately, I liked phase because I imagined the beginning of multiphasic technologies like the science fiction way Borg shields would respond to different types of weapons in the Star Trek universe. Their shields were multiphasic and adapted to the frequencies and phase-shifts of different weapons like phasers. But I was certainly very open to the use of mode and state. It turned out, Ray had a sort of preference for phase as well, but mostly because he was curious about developing a spatial form of phase dependent on the electrostatic configurations of electrons in my work that gave a classical connection to the periodic table of elements. In my mind, this was always an embodiment of entropy and helped shape some of my broader thinking about entropy as part of a symmetry function or functional that we'd develop to inform quantum mechanics and density functional theory. That work hasn't yet emerged by the way.

So, in his book, Ray used the term "Intra-phasic" on page 135 where he singles out my work in three brief paragraphs. He capitalized it as though it were a proper noun as he does several other common nouns in his writing, and he hyphenates it perhaps as he's not quite ready to settle upon the singular work intraphasic as part of his lexicon. Maybe his capitalization is meant to draw attention to the term and the hyphenation is his way of emphasizing the root phase that he often mentioned as we later worked together. Either way, I believe all these ideas were at work when he wrote this down.

What does intraphasic mean? Well, in this case, he's describing intraphasic energy, and in particular, electrostatic potential energy in a given $N$-electron system. The energy difference he points to, that which I discovered corresponds remarkably well with physical properties of atomic electron structure throughout the periodic table(!), is $E(N-1,1e)-E(N)$. In my own work since my college days, I've been more careful about notation (my undergraduate electromagnetics professor, Dr. Thomas Erber often emphasized "notation, notation, notation!" when describing the three most important things in science just as "location, location, location" are the three most important things in real estate. I'm not sure if this was his invention, or Niels Bohr's since Erber's hero was Bohr.), so I use $U(N)$ instead of $E(N)$ to exclusively denote potential energy rather than the total energy (to include kinetic), but the idea he wants to convey is the same. There's an energy difference.

In this special case, we're referring to the $N$ electrons inside a spherical dielectric object (with linear electric response) like a spherical quantum dot. $E(N)$ is the total electrostatic potential energy of a given $N$-electron solution of the system in which case, all electrons are found to reside at approximately the same distance from the origin of the sphere, but still inside the sphere. Consider the $N=4$ electron solution as a simple gedanken experiment of your own (I'll post some pretty pictures here later for the imagination-impaired). The electron configuration that minimizes the total global electrostatic potential energy of a 4-electron system is that when electrons reside at the vertices of a regular tetrahedron centered about the origin. Next, consider not only what happens to this tetrahedral geometric symmetry but also consider how much energy is needed to move one of these four electrons to the origin of the sphere. In this process, energy is added to the system because this new configuration is not the electrostatic global minimum (or absolute ground state) configuration (the tetrahedron is the lowest possible energy configuration). So with $E(N-1,1e)$ being the energy of this new configuration ($1e$ represents the separated electron at the origin and $N-1$ represent the remaining electrons surrounding it) the energy difference, $E(N-1,1e)-E(N)$, is always positive. As well, the remaining $N-1$ electrons now reside at vertices of an equilateral triangle centered about the origin -- the location of the displaced electron. Thus, there is a change of energy due to a discrete change in the geometric symmetry (from tetrahedron to triangle) of the system while the volume of the dielectric sphere and the number of electrons did not change. In more mathematical terms, one generally has energy differences that depend on changes in volume (size), number of electrons, and geometric symmetry (which again, I prefer to use the term entropy). Here, I devised a way to exclusively consider the energy difference due to geometric symmetry changes. Hence, this energy difference is an intraphasic energy of the system in which nothing changes but the electron configuration (the so-called virial). It's an internal (intra-) change.

I was happy to see this terminology in his book published in 2018.

Other variations of this root included interphasic, multiphasic, electrophasic, and monophasic. Monophasic was perhaps used more readily than the others because this, after all, is electrostatics. It represents the spatial phase of a particular $N$-charge system. Hence, monophasic capacitance is a unique terminology we applied in some of our published papers in which I defined the capacitance of nanoscale, few-electron systems by an average electric electric potential (equipotential) experienced by all electrons in the system. In fact, it is likely shocking to many electrical engineers to realize that an equipotential surface is exceptionally rare in general and that the reason why our mathematical framework for electronic circuit elements is historically pretty simple because the metallic interconnects firmly pin-down the shape of equipotential surfaces throughout the system. In general, we don't have metal interconnects though, and the smaller our devices become, the fewer charged particles (e.g. electrons) there are to define the electric potential landscape.

My favorite of these new terms is electrophasic. It's not one that I've used it in publications, or I don't recall doing so. Electrophasic capacitance, for example is the capacitance in the volume of space surrounding a single electron within an electrostatic configuration of the system -- the monophasic system. I challenge you the reader to think about how you would sum up all the electrophasic capacitances of a 5-electron monophasic dielectric sphere. To get you started, the geometric solution occurs when two electrons reside at the north and south poles and the remaining three form vertices of an equilateral triangle about the equator. Taken together, they form a triangular bipyramid centered about the origin. You'll quickly see why there is no such thing as a global equipotential experienced by all 5 electrons, that the concept of parallel and series capacitances is therefore insufficient and that we either need something new to define the total capacitance of this system or we can settle for the average electric potential experienced by all 5 electrons as the potential reference in our calculation. (Infinite distance from the system may be our ground potential.) If we choose the latter, notice that as the number of electrons becomes very large (tending toward a spherical metal shell of energy $N^2/2$) we recover an exceptionally great approximation of an equipotential surface, but that in the case of $N=1$ the capacitance differs from the textbook expression by a factor of $1/2$. Notably, the textbook expression is dependent on the assumption of an equipotential surface that's historically pinned-down by the presence of a metal connect or plate. In short, our textbooks do not present a general expression of capacitance.

I liked the idea of an interphasic system because this allows us to discuss what happens when a system gains or loses an electron. Not only does the virial change (the geometric symmetry) but so too does $N$. We can even begin to think about how two neighboring systems exchange or share an electron similar in context to ionic or covalent bonding of atoms, respectively. In the case of modeling within a dielectric environment, think about how an electron might be shared if the region of space between two spheres exhibits sufficient dielectric magnitude that an electron could reside within this space with less electrophasic energy than at any location within either of the two original spheres. Yes, we'll now need to break away from the notion of a dielectric constant and begin to discuss a spatially-dependent dielectric function. In the limit, I'd love to know how the dielectric function varies in the space between a proton and an electron. Is that the simplest possible case of a dielectric function?

Multiphasic properties emerge in this case when we begin to think about neighboring systems containing different values of $N$ and therefore entirely different virials that ultimately bond in one manner or another. Do we recover periodic Bloch functions, for example, in crystal lattices?

. . .just several of many new tunnels to explore in this rabbit hole I stumbled upon in grad school that may lead to very familiar science concepts, laws, and models.