1. Introduction

https://doi.org/10.48550/arXiv.2403.13981

Most of the classical electromagnetic theory that is commonly described in textbooks was established in the 19th century before electrons had been discovered or the existence of atoms had been confirmed [Maxwell, 1865; Maxwell, 1873; Maxwell, 1892; Heaviside, 1893; Lorentz, 1916; Buchwald and Fox, 2013]. The constitutive relations, ${\D = \varepsilon_0 \E + \pp}$ and ${\hh= \mu_0^{-1}\B - \M}$, between the macroscopic electric and magnetic fields, ${\E}$ and ${\B}$, the induced fields ${\pp}$ and ${\M}$, and the auxiliary fields ${\D}$ and ${\hh}$, are an important part of this theory. To deduce them, materials were approximated as continua at the macroscale; and the polarization ($\pp$) and magnetization ($\M$) densities were introduced to characterize how the state of the aether was altered by their presence. When the concept of an aether was abandoned, $\pp$ and $\M$ were reinterpreted as linear electromagnetic responses of materials. However this appears to have been done ad hoc and without due concern for consistency with the nascent theory of material microstructure.

The microscopic theory or vacuum theory of electromagnetism rightly underpins our microscopic theory of material structure, composition, and energetics. The purpose of macroscopic electromagnetism, which reduces to microscopic electromagnetism when ${\pp}$ and ${\M}$ vanish, is to provide a unified description of materials and electromagnetic fields at the macroscale. Therefore it should be underpinned by our mutually-consistent microscopic theories of materials physics and vacuum electromagnetism; and we should understand the microscopic origins of ${\pp}$ and ${\M}$ clearly. However development of the macroscopic theory was completed before many of the discoveries on which we base our microscopic understanding of materials were made, and it quickly became an established part of physics doctrine. Therefore it was not built on firm microscopic foundations and, unfortunately, it has never been reconciled fully and satisfactorily with microscopic physics.

Inconsistencies between the microscopic and macroscopic theories were not apparent to most scientists until after crystallography had come of age and it had become possible to compute materials’ microstructures, by which I mean the statistical distributions, on the nanoscale, of their constituent charges and magnetic moments. It became obvious that we lacked precise and viable definitions of ${\pp}$ and ${\M}$ when attempts were made to define them in terms of microstructures [Martin, 1974; Littlewood and Heine, 1979; Littlewood, 1980; Vogl, 1978; Tagantsev, 1991; Resta, 1992; Aizu, 1962; Landauer, 1981; Landauer, 1960; Larmor, 1921; Woo, 1971]. Neither $\pp$ nor $\M$ is directly measureable, but definitions of observables attributed by classical electromagnetic theory to changes in their values also proved elusive. For example, it was not until the 1990s that researchers discovered how to calculate the so-called polarization current, $\Jconv$, that flows through an insulating inversion-asymmetric crystal when it is uniformly perturbed by a stimulus, such as a strain or a change in temperature. It is $\Jconv$, and not ${\pp}$ or ${\D}$, that is measured directly in experiments that produce ${\pp-\E}$ or ${\D-\E}$ hysteresis loops.

An exact calculable expression for $\Jconv$ was eventually provided by a theory that became known as the Modern Theory of Polarization (MTOP) [Resta, 1993; King-Smith and Vanderbilt, 1993; Vanderbilt and King-Smith, 1993; Resta, 1994; Resta and Vanderbilt, 2007; Resta, 2010]. According to the MTOP, a crystal’s bulk polarization $\pp$ is a multivalued quantity that cannot be calculated from the crystal’s microscopic charge density $\rho$, because it is a property of the phase of the crystal’s wavefunction, which has ‘nothing to do with’ the charge density [Resta, 2018; Resta, 2010; Resta, 1993]. Therefore the MTOP deviates substantially from both the revised (20th century) definition of $\pp$ as a dipole moment density, and the physical reasoning with which the use of $\pp$ as a measure of dielectric response to a uniform field was justified.

Another elusive and hotly-debated definition was that of the areal density of charge at a surface or interface, ${\bsigma[\rho]}$. This problem was solved for crystals by Finnis in 1998 [Finnis, 1998], and there appears to be agreement now that his definition is correct [Resta and Vanderbilt, 2007; Goniakowski et al., 2008; Stengel and Vanderbilt, 2009; Stengel, 2011; Bristowe et al., 2011; Noguera and Goniakowski, 2013; Goniakowski and Noguera, 2014; Bristowe et al., 2014; Vanderbilt, 2018]. Unfortunately, the literature is far from clear on this point because multiple equivalent definitions of ${\bsigma[\rho]}$ have been proposed, and some works continue to make the unnecessary distinction between free charge and bound charge, and to express the latter’s contribution to $\bsigma$ as ${\bsigmabound=\pp\cdot\normal}$, where ${\normal}$ is the unit surface normal and $\pp$ is the polarization in the crystal’s bulk. The definition is complicated further by the multivaluedness of the MTOP definition of ${\pp}$.

A third illustration of the tension that exists between the 20th century theory of material structure and 19th century electromagnetism, is the question of how to define and calculate the bulk macroscopic electric potential, $\bphi$, from the microscopic charge density, $\rho$. This quantity, which is often called the mean inner potential (MIP), plays an important role in several areas of research, including theoretical electrochemistry and electron microscopy [Miyake, 1940; Sanchez and Ochando, 1985; Wilson et al., 1987; Wilson et al., 1988; Wilson et al., 1989; Pratt, 1992; Gajdardziska-Josifovska et al., 1993; Spence, 1993; Sokhan and Tildesley; Spence, 1999; Leung, 2010; Kathmann et al., 2011; Cendagorta and Ichiye, 2015; Blumenthal et al., 2017; Hörmann et al., 2019; Yesibolati et al., 2020; Madsen et al., 2021; Kathmann, 2021; Peng, 1999; Saldin and Spence, 1994; Gajdardziska-Josifovska and Carim, 1999; Ibers, 1958; Rez et al., 1994; Sokhan and Tildesley; Stillinger and Ben‐Naim, 1967]. An exact general definition of it has not previously been found, but several approximations to it have been proposed and are in use [Kathmann et al., 2011; Sokhan and Tildesley; Saunders et al., 1992; Pratt, 1992]. Bethe derived one such expression by approximating the microstructure as a superposition of spherically-symmetric atomic charge densities [Bethe, 1928].

1.1 Motivations and objectives

An electrical microstate, ${(\micro,\dmicro)}$, is an electrical microstructure $\micro$ and its time derivative, ${\dmicro\equiv\partial \micro/\partial t}$, at the same instant. An electrical microstructure is the most complete and detailed information pertaining to the instantaneous spatial distribution of charge and electric potential that could, in principle, exist. It could be specified by the wavefunction, density matrix, or position probability density function (pdf) of the set of all particles, but for many purposes the information required is contained in the microscopic electric potential $\phi$ and the microscopic charge density ${\rho\equiv-\laplacian\phi}$, in which case we say that the electrical microstate is ${(\phi,\dphi)}$ or ${(\rho,\drho)}$. An electrical macrostate, ${(\macro,\dmacro)}$, is a specification of the spatial distributions of charge and electric potential at the macroscale, $\macro$, and their time derivatives, $\dmacro$.

In each of the three examples discussed above, a different line of reasoning was followed to derive an expression for ${\Jconv=\Jconv[\dmicro]}$, ${\bsigma=\bsigma[\micro]}$, or ${\bphi=\bphi[\micro]}$. However, none of these lines of reasoning were pursued far enough to elucidate the relationship between macrostructure and microstructure fully, and with enough generality that what was learned could be applied, not only throughout electromagnetic theory, but far beyond it: in elasticity theory, meteorology, astrophysics, and countless other areas of research. For example, the MTOP did not provide an expression for ${\bphi[\micro]}$, and Finnis did not derive an expression for ${\Jconv[\dmicro]}$ from his expression for ${\bsigma[\micro]}$.

1.1.1 Objective 1

I use the term macroscopic field to mean uniform field. A uniform field can be regarded as a field whose wavelength is orders of magnitude larger than the material to which it is applied or from which it emanates. I say little about fields whose wavelengths are shorter than, or comparable to, a material’s linear dimensions. The physics of such fields is qualitatively different, in some respects, to the physics of macroscopic fields.

1.1.2 Objective 2

If you look around you, you will see surfaces, edges, and corners everywhere. Everything you see is a feature of the macrostructure, meaning that it is a blurred image of the microstructure at a surface, edge, or corner. You do not see the microstructure in its full horrendous complexity, and you do not notice that it changes from one femtosecond to the next. You see a relatively simple and relatively stable homogenized version of the microstructure.

There are many sources of imprecision, such as the diffraction limit, and it would be impossible for you to be aware of the full microstructure because, for example, a cubic molar sample of an element has ${\sim 10^{16}}$ atoms at each of its six faces, but the human brain only has ${\sim 10^{11}}$ neurons. Therefore homogenization of microstructure to form macrostructure is intrinsic to the act of observation.

Nevertheless, given a microstructure $\nu$ and access to an arbitrarily-powerful computer, it is not known how to calculate the macrostructure, or even what mathematical form it would take. The inconsistencies between Maxwell’s macroscopic and microscopic theories of electromagnetism are only one of many important consequences of this gap in our understanding.

Therefore I address the following question, which is of general importance to mathematical physics:

How can a macrostructure be expressed mathematically in terms of the microstructure underlying it?

This question leads quickly to a more fundamental question:

If the microstructure is a scalar field ${\nu:\realone^3\to \realone}$, what is the mathematical form of the macrostructure?

It turns out (see Sec. 8) that qualitative differences exist between a macrostructure and a base microstructure, where I use the term base microstructure to mean a microstructure that is not itself the macrostructure arising from a structure on an even smaller length scale.

1.1.3 Objective 3

Most textbooks on solid state physics or electronic structure theory do not clearly demarcate the features of mathematical representations of statistical microstates that are peculiar to quantum mechanics for fundamental physical reasons, from features that are consistent with classical statistical microstates. For example, when we see a statistical state expressed as ${\Psi\equiv\sqrt{\pdf}e^{i\theta}}$, where ${\pdf=\pdf(\rvec_1,\rvec_2,\cdots)}$ is a position pdf, we often assume that quantum mechanics is being ‘used’. However, it is perfectly valid to express the statistical state of a system of classical particles in this form, and it can be useful to do so. Having done so, the classical many-particle state $\Psi$ can be expanded in a basis of single particle states, just as in quantum mechanics.

We largely base our physical intuitions on what we observe at the human scale. If the blurred lines between classical and quantum physics were made more clear, we would have a better understanding of when we could apply our classical intuitions to systems of quantum mechanical particles, and when our intuitions were likely to fail us.

I begin to address this issue in the present work for two reasons. The first is that fulfilling my first objective, and relating my findings to the MTOP, requires me to survey many parts of electronic structure theory and solid state physics. Therefore I have the opportunity to point out that much of the mathematical infrastructure that we usually associate with quantum mechanics is perfectly consistent with classical physics.

The second reason is that there are claims in the literature on the MTOP that some of the observable quantities that I discuss in this work have quantum mechanical origins and do not have analogues within classical physics [Resta, 1994; Resta, 1993]. It is important to examine these claims carefully.

Single particle states play a prominent role in the MTOP. Therefore I emphasize that there is nothing specific to quantum mechanics about Bloch functions [Bloch, 1929] and Wannier functions [Wannier, 1937]. If the bulk of a crystal is represented in a torus, which is equivalent to using Born-von Kármán boundary conditions [Born and von Kármán, 1912], and if $\Psi$ is a stationary statistical state resulting from a classical process that preserves the crystal’s periodicity, it can be expanded in a basis of Bloch functions. Each set of Bloch functions can be transformed into an infinite number of sets of Wannier functions, which must include a maximally localized set [Ferreira and Parada, 1970; Marzari and Vanderbilt, 1997].

The MTOP approach to calculating $\Jconv$ gives exactly the right result when the charge density can be expressed in the form

The MTOP is routinely applied to electrons in insulators by representing their electron densities as sets of smoothly-evolving Bloch or Wannier states of fixed occupancies; and it is trivial to apply it to classical nuclei or ions if they can be treated as point charges whose positions as functions of $\zeta$ are known. However it is probably not known how to adapt and apply the MTOP to the ${\zeta}$-dependent statistical state of an arbitrary classical or quantum mechanical physical system. I do not shed light on the answer to this representability problem, but I attempt to clarify the question and to emphasize its importance.

1.2 Theoretical approach and outline of this work

I define the homogenization transformation that turns an electrical microstructure into an electrical macrostructure as a spatial averaging operation on a mesoscopic domain. This obvious approach, which appears physically reasonable, has been attempted many times before by many authors [Kaufman, 1961; Rosenfeld, 1965; de Groot and Vlieger, 1965; Schram, 1960; Russakoff, 1970; Robinson, 1971; de Groot and Vlieger, 1964; Mazur and Nijboer, 1953; Kirkwood, 1936; Mazur, 1957; Frias and Smolyakov, 2012; de Lange et al., 2012; Raab and de Lange, 2005; de Lange and Raab, 2006; Roche, 2000]; and some of these attempts are presented in well known textbooks [Jackson, 1998; Ashcroft and Mermin, 1976]. However none of these approaches have been adopted widely by the research community as foundations for the development of rigorous theory, because they do not lead to Maxwell’s macroscopic theory of electricity.

In Sec. 4 I explain why we should not be deterred by this: I outline the reasoning that led Maxwell to his macroscopic theory in order to demonstrate that this reasoning has been invalidated by what has since been learned about spacetime and the microstructures of materials. Therefore we should not require unobservable elements of Maxwell’s macroscopic theory, such as $\pp$, to be elements of a macroscopic theory that is derived from, and consistent with, his vacuum theory of electromagnetism

In Sec. 5, using the macroscopic polarization $\pp$ as an example, I briefly explain some of the ways in which definitions of macroscopic fields have failed in the past.

In Sec. 6 I argue that many of my conclusions, and many elementary aspects of electricity at the macroscale, are demands of symmetry or asymmetry.

For example, any stimulus changes the microscopic charge density in the bulk of a crystal, to some degree. While $\rho$ is changing, microscopic polarization current ($\jconv$) flows, because ${\partial \rho/\partial t=-\div\jconv}$. Whether or not a net, or macroscopic, polarization current ($\Jconv$) flows depends on the symmetry of the composite crystal+stimulus system: The component of $\Jconv$ in direction $\uhat$ vanishes if there is a glide plane normal to $\uhat$, because then the sum,

As another example, suppose that the average of microscopic charge density ${\rho_k(x,y,z)}$ over the Cartesian ${y-z}$ plane is

In Sec. 7 I discuss the Modern Theory of Polarization, and I derive the MTOP expression for ${\Jconv}$ without invoking quantum mechanics.

In Sec. 8 I explain in more detail what I mean by the prefixes micro- and macro-. I outline some of the qualitative differences between a macrostructure and a base microstructure and I explain the mathematical and physical origins of those differences.

In Sec. 9 I discuss the macroscopic excess fields that exist at surfaces, interfaces, edges, and line and point defects. Excess fields are the manifestations at the macroscale of abrupt changes of the microstructure, meaning changes that occur across microscopic distances. For example, the difference in microstructure between a material and vacuum manifests as an areal charge density $\bsigma$ on the material’s surface. I derive expressions for macroscopic excess fields in terms of microscopic volumetric fields, which generalize Finnis’s expression for surface excesses to non-periodic microstructures.

In Sec. 10 I use spatial averaging of the microscopic charge density $\rho$ to calculate its macroscopic counterpart $\Rho$, and in Sec. 12 I define the surface charge density $\bsigma$ as the integral of $\Rho$ along a path that crosses the surface. This leads to Finnis’s expression for the surface charge of a crystal, ${\bsigma[\rho]}$, and to my generalization of this expression to noncrystalline materials.

In Sec. 13 I derive the MTOP expression for $\Jconv$ again, but this time I derive it by defining it as ${\Jconv\equiv\dbsigma=\mathrm{d}\bsigma[\rho]/\mathrm{d}t}$, where ${\bsigma[\rho]}$ is Finnis’s formula.

In Sec. 14 I point out that there does not exist a theoretical justification for interpreting the sets of single electron states that appear in the MTOP definitions of polarization current as chemically meaningful substructures of the electron density.

In Sec. 15 I prove that the macroscopic potential, or mean inner potential, $\bphi$ vanishes in the bulk of any isolated material whose surface is locally charge neutral. Then I point out a flaw in the reasoning used by H. A. Lorentz to deduce that a macroscopic electric field exists in the bulk of a crystal whose microstructure is anisotropic. In Sec. 16 I discuss flaws in Bethe’s derivation of an approximate expression for $\bphi$, and I show that $\bphi$ vanishes when these flaws are avoided.

This work comprises three interwoven strands, whose individual objectives are the three objectives outlined above. It concludes in Sec. 17 with a summary of each strand.

1.3 Selectively reading this work

Those interested only in the homogenization transformation that turns microstructure into macrostructure, should read Sec. 8 and Sec. 9, and Appendix J. Those interested in everything except the homogenization transformation, and who are willing to trust the formulae derived in Sec. 9 and presented in Appendix A, can safely skip those parts.

Those interested only in the mean inner potential, $\bphi$, should read Sec. 6 (particularly Sec. 6.6), Sec. 15 and Sec. 16.

Those interested only in polarization current, $\Jconv$, or the Modern Theory of Polarization should read Sec. 6, Sec. 7, Sec. 12, and Sec. 13.

Those interested only in the macroscopic electric field, $\E$, should read Sec. 6 and Sec. 15.

Those interested only in surface charge, $\bsigma$, should read Sec. 8.3, Sec. 9 and Sec. 12.

Those interested only in single particle states should read Sec. 7.5, Sec. 13.2.1, and Sec. 14, and Appendices Appendix G, Appendix H, and Appendix I.

Those interested only in the relationship between quantum mechanics and classical statistical mechanics should read Sec. 3.4 and Sec. 14.2.7, and Appendices Appendix C, Appendix D, Appendix E and Appendix F.

I would be grateful for critical feedback on any part or aspect of this work (p.tangney@imperial.ac.uk).