Appendix A. Areal charge densities and other integrals of macroscopic fields across interfaces

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

Here I quote expressions for the excesses of microscopic scalar fields at macroscale surfaces and interfaces. Derivations of these expressions can be found in Sec. 9 and a derivation of a special case of one of them can be found in [Finnis, 1998].

I use the microscopic charge density ${\rho}$ as my example, but the expressions are applicable to any scalar field $\nu$, including scalar components of vector or tensor fields. They are valid when there exists a mesoscale $l$ on which the statistical characteristics of the microscopic ($\sim a$) fluctuations in $\nu$ do not vary appreciably and the macroscale counterpart ${\Nu}$ of ${\nu}$ varies at most linearly. When that is the case it is possible to calculate a mesoscale average ${\bnu}$ of ${\nu}$ at each point, such that ${\Delta\nu\equiv \nu-\bnu}$ fluctuates microscopically about zero, and such that ${\bnu}$ equals ${\Nu}$ to within the finite precision ${\precNu}$ to which ${\Nu}$ is defined.

Since ${\rho}$ fluctuates microscopically about zero in a material’s bulk, ${\brho=0}$ is among the values of its mesoscale average at each point. Therefore, I define ${\Delta\rho\equiv\rho}$ and I will express the quoted formulae in terms of ${\rho}$, rather than ${\Delta \rho}$. Before quoting them, it is necessary to explain the construction used to define the quantities appearing within them.

Consider a material with two surfaces which, on the mesoscale, are locally planar where they intersect the Cartesian $x$ axis. Let ${x_L}$ and ${x_R}$ be points on the $x$ axis in the vacuum immediately outside the material at the surfaces with outward unit normals ${-\hat{x}}$ and ${+\hat{x}}$, respectively. Let ${x_b\in (x_L,x_R)}$ be a point on the $x$ axis that is arbitrary apart from the requirement that it is far enough away from both surfaces that it can be regarded as being in the bulk of the material. In each of the expressions quoted below, ${x_b}$ should be regarded as a point in the bulk below whichever surface (at $x_L$ or $x_R$) the expression pertains to.

Now consider a mesoscopic neighbourhood ${\interval(x_b+u,\ell)}$ of $x_b$, where ${u\sim a}$ and ${\ell\sim l}$. Let us assume that it is partitioned into contiguous microscopic (${\sim a}$) intervals ${\interval(\bar{x}_m,\Delta_m)}$, such that ${x_b}$ is at the boundary point shared by two of them, and such that the average of ${\rho}$ on each interval vanishes. Let ${\manyrho(\bar{x}_m,\Delta_m)\equiv \Delta_m^{-1}\int_{-\Delta_m/2}^{\Delta_m/2}\rho(\bar{x}_m+u)\,u^n\,\dd{u}}$ be the $n^\text{th}$ moment of the ${m^\text{th}}$ interval divided by its width ${\Delta_m\sim a}$, and let $\bmanyrho(x_b)$ denote the average of ${\manyrho(\bar{x}_m,\Delta_m)}$ over all intervals in the set that partitions ${\interval(x_b+u,\ell)}$.

Then, if ${\Rho}$ denotes the macroscopic counterpart of ${\rho}$, and ${\mxl}$ and ${\mxr}$ denote macroscale points in the vacuum beyond the surfaces normal to ${-\hat{x}}$ and ${\hat{x}}$, respectively, the areal densities of charge at these surfaces can be expressed as

\begin{align*} \bsigma_L & = \int_{\mxl}^{\mxb}\Rho(\bx)\dbx \\ &= \int_{x_L}^{x_b}\rho(x) \dd{x} - \bmonerho(x_b), \tag{99} \\ \bsigma_R &= \int_{\mxb}^{\mxr}\Rho(\bx)\dbx \\ &= -\int_{x_b}^{x_R}\rho(x) \dd{x} + \bmonerho(x_b), \tag{100} \end{align*}
and the integrals of ${\bx\Rho(\bx)}$ across the surfaces can be expressed as
\begin{align*} \int_{\mxl}^{\mxb} \bx \Rho(\bx)\dbx = \int_{x_L}^{x_b} x \rho(x)\dd{x} &- x_b\bmonerho(x_b) \\ &-\bmtworho(x_b) \tag{101} \\ \int_{\mxb}^{\mxr} \bx \Rho(\bx)\dbx = \int_{x_b}^{x_r} x \rho(x)\dd{x} &+ x_b\bmonerho(x_b) \\ &+\bmtworho(x_b) \tag{102} \end{align*}
The expression for the charge $\bsigma$ at an interface between two materials can be deduced from these expressions by assuming that ${\mxbl}$ and ${x_{bL}}$ are macroscale and microscale points, respectively, in the bulk of one of the materials, and that ${\mxbr>\mxbl}$ and ${x_{bR}>x_{bL}}$ are points in the bulk of the other.
\begin{align*} \bsigma &= \int_{\mxbl}^{\mxbr} \Rho(\bx)\dbx \\ &= \int_{x_{bL}}^{x_{bR}}\rho(x)\dd{x} + \bmonerho(x_{bL}) -\bmonerho(x_{bR}) \tag{103} \end{align*}
It follows from this expression that the excess of charge at any plane in the bulk of a macroscopically-uniform material vanishes. The plane can be treated as an interface and, since the plane itself is in the bulk, the values of ${x_{bL}}$ and ${x_{bR}}$ can both be chosen to be the point at which the $x$ axis intersects the plane. Then the right hand side of Eq. (103) vanishes.

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