5. How is $\pp$ defined?

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

Many attempts have been made to reconcile Maxwell’s auxiliary electric fields, $\pp$ and $\D$, with modern conceptions of material structure and composition. Most have proposed definitions of $\pp$ in terms of the microscopic charge density $\rho$. Once definitions of $\E$ and $\pp$ are in hand, the definition of $\D$ follows from the constitutive relation ${\D=\varepsilon_0\E+\pp}$.

None of the proposed definitions of $\pp$ are viable, to my knowledge, and I briefly explain their shortcomings in this section. I do not attempt to refute every paper directly, but I outline a few of the most common definitions of $\pp$ and the reasons why they are unsatisfactory.

The literature on the Modern Theory of Polarization (MTOP) (e.g., [Resta and Vanderbilt, 2007]), which is discussed in Sec. 7, can be consulted for more discussion about shortcomings of pre-MTOP definitions of $\pp$. I outline reasons to look beyond the MTOP definition of $\pp$ in Sec. 7.

5.1 Attempt 1

$\pp$ has the dimensions of a dipole moment per unit volume and so it is natural to try to define it as such. Let us consider a material that occupies and fills a space ${\material\subset\realthree}$, whose volume is ${\materialvolume}$. Many authors have assumed, often tacitly, that $\material$ can be divided into microscopic partitions in some natural or ‘right’ way. For example, in a molecular material there might be a separate partition for each molecule. $\pp$ is then defined as the macroscopic spatial average of the partitions’ dipole moments divided by their volumes. This definition fails because, as Fig. 11 and Fig. 12 illustrate, there are an infinite number of ways to partition any material, which are equally justifiable theoretically, and each different set of partitions leads to a different magnitude and direction of $\pp$, in general.

5.2 Attempt 2

One could also define $\pp$ as the dipole moment of the entire material divided by its volume, i.e.,
\begin{align*} \pp\equiv \frac{1}{\materialvolume}\int_\material \rho(\rvec)\,\rvec\ddpow{3}{r}. \tag{5} \end{align*}
This definition is not satisfactory because it implies that $\pp$ is not a property of the bulk, in general. To understand why, consider the dipole moment of a charge-neutral crystalline rod, of length $L$ and area of cross-section $A$, whose surfaces perpendicular to its long axis and carry charges of $+q$ and $-q$. For simplicity let us suppose that the rod is carved from a perfect ionic crystal, and then isolated immediately so that its composition cannot change.

If $L$ was large compared to the crystal’s lattice constant, the rod’s dipole moment would be approximately equal to ${qL\,\normal}$, where $\normal$ is an outward unit normal to the surface that carries charge $q$. However, both the magnitude and the sign of $q$ are determined by where along the rod’s axis the bulk crystal was cleaved to form its surfaces. As illustrated in Fig. 12, two surfaces formed by cleaving a crystal along relatively-shifted parallel planes have different charges, in general. Therefore, by this definition, the value of

\begin{align*} \pp\approx \frac{qL}{\materialvolume}\,\normal= \frac{q}{A}\,\normal \end{align*}
is mostly determined by the areal density of charge on the rod’s surfaces, ${\bsigma\equiv q/A}$, and it would vanish if the surfaces were neutralized. Therefore, Eq. (5) does not define a bulk property.

5.3 Attempt 3

One could consider basing a definition on either ${-\div\pp=\Rho}$ or ${-\div\mpp=\rho}$, where $\mpp$ is a microscale analogue of $\pp$. This approach fails because, without boundary conditions, these equations only define $\pp$ and $\mpp$ up to arbitrary constants. With boundary conditions, their values are determined by the charge at the material’s surfaces, so they are not properties of the bulk.

5.4 Attempt 4

Finally, several well-known textbooks, including those by Jackson [Jackson, 1998] and Ashcroft and Mermin [Ashcroft and Mermin, 1976], use variants of a method refined by Russakoff [Russakoff, 1970] to define $\pp$. They assume that the microscopic charge density can be expressed in the form
\begin{align*} \rho(\rvec) = \sum_i\rho_i (\rvec-\rvecsub{i}), \end{align*}
where each $\rho_i$ is a charge density that is localized around the origin and microscopic in extent, such that ${\rho_i(\rvec)\approx 0}$ when ${\abs{\rvec}\gg a}$. For example, each $\rho_i$ might be the distribution of a different molecule’s charge.

The next step is to find the spatial average of $\rho$ by convolving it with a smooth spherically-symmetric averaging kernel, ${\mu(\epsilon):\realthree\to\realnonneg}$, whose width is proportional to ${\epsilon}$ and which has an integral of one, i.e.,

\begin{align*} \expval{\rho;\mu}_\epsilon(\rvec) &\equiv \intthree \mu(\uvec;\epsilon) \, \rho(\rvec+\uvec)\ddpow{3}{u} \\ & = \sum_i \intthree \mu(\rvec-\rvecsub{i}-\uvec;\epsilon) \, \rho_i(\uvec)\ddpow{3}{u}, \end{align*}
where I have changed the variable of integration and made use of $\mu$’s spherical symmetry. Taylor expanding ${\mu(\epsilon)}$ in each integrand to first order in $\uvec$ gives
\begin{align*} \expval{\,\rho\,}(\rvec) & \approx \sum_i \intthree \left[\mu(\rvec-\rvecsub{i};\epsilon) - \uvec\cdot\grad \mu(\rvec-\rvecsub{i};\epsilon)\right] \\ &\hspace{2cm} \times\rho_i(\uvec)\ddpow{3}{u} \\ & = \sum_i \mu(\rvec-\rvecsub{i};\epsilon)\,q_i - \sum_i\grad \mu(\rvec-\rvecsub{i};\epsilon)\cdot\dvecsub{i} \end{align*}
where ${q_i\equiv\intthree \rho_i(\rvec)\ddpow{3}{r}}$ and ${\dvecsub{i}\equiv\intthree \rho_i(\rvec+\uvec)\,\uvec\,\ddpow{3}{u}}$ are the net charge and the dipole moment of distribution $\rho_i$, respectively.

If we identify ${\expval{\rho;\mu}_\epsilon}$ as the macroscopic charge density $\Rho$, and express the second sum on the right hand side as

\begin{align*} \div\left(\sum_i \mu(\rvec-\rvecsub{i};\epsilon)\,\dvecsub{i}\right), \end{align*}
we find that
\begin{align*} \Rho & = \Rhofree - \div{\pp} = \Rhofree + \Rhobound, \end{align*}
where ${\Rhobound\equiv-\div\pp}$; and $\Rhofree$ and $\pp$ are volumetric densities of the molecules’ net charges and dipole moments, respectively.

There are multiple fatal flaws in these definitions of $\Rhobound$, $\Rhofree$, and $\pp$: The value of each quantity depends sensitively on the value of $\epsilon$, which is arbitrary; and both $\Rhobound$ and $\pp$ vanish in the limit ${\epsilon\to\infty}$. Each field also depends on how $\rho$ is partitioned into localized distributions ${\rho_i}$. However, even if the set ${\{\rho_i\}}$ was given, if ${\Rhofree\neq 0}$ then the values of $\Rhobound$, $\Rhofree$, and $\pp$ would depend on the choice of origin. This is because the dipole moment of any charge distribution is origin-dependent, unless its net charge is zero.

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