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The electromagnetic stress tensor in cubic crystals and amorphous solids

This paper derives the electromagnetic vacuum stress tensor for cubic crystals and amorphous solids by demonstrating its agreement between microscopic and phenomenological approaches, revealing a nontrivial constant unattainable via macroscopic electrodynamics, and showing that the transverse component in amorphous systems matches Peierls' result for liquids.

Original authors: Richard Dengler

Published 2026-09-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Richard Dengler

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light and electricity are not just waves that pass through matter; they also push on it. This push, known as stress, is a fundamental concept in physics that describes how electromagnetic fields transfer momentum to the material they traverse. In the macroscopic world, where we deal with bulk materials like glass or water, scientists have long used a set of rules called macroscopic electrodynamics to predict these forces. These rules work remarkably well for many things, such as calculating how much light bends when it enters a lens or how a capacitor stores energy. However, when it comes to the specific pressure exerted by light inside a solid, the standard rules have left physicists in a state of debate for decades. The difficulty lies in the fact that the standard formulas rely on a property of the material—how its ability to store electric energy changes when the material is squeezed or stretched—that is often unknown or impossible to measure directly. Without this missing piece, the standard theory cannot definitively predict the internal stress of a solid, leading to questions about whether the macroscopic rules are sufficient on their own or if they are missing something fundamental about the atomic structure of matter.

Richard Dengler, working from Munich, has tackled this problem by looking at the material not as a smooth, continuous block, but as a collection of individual atoms arranged in a precise grid. He modeled a crystal as a lattice of tiny, classical atoms, each acting like a small dipole that can be polarized by an electric field. By calculating the electric field generated by these discrete atoms and summing up the forces they exert on one another, he derived the stress tensor—the mathematical description of the internal pressure—from the ground up. This microscopic approach allowed him to calculate the exact pressure in two specific directions: one parallel to the electric field and one perpendicular to it. The results were striking. The calculation revealed that the stress tensor contains a specific, non-zero constant that depends entirely on the microscopic arrangement of the atoms. This constant means that the pressure inside the crystal cannot be derived solely from the macroscopic properties of the material; the standard equations, which ignore the atomic lattice, are incomplete. The study confirms that to truly understand the forces at play, one must account for the discrete, grainy nature of matter at the atomic scale.

To make the findings more broadly applicable, the researcher then applied this logic to amorphous solids, which are materials like glass that lack a regular crystal structure. He modeled an amorphous solid as a mosaic of tiny, randomly oriented grains of cubic crystals. By averaging the stress calculated for the individual crystals over all possible directions, he determined the overall stress in the disordered material. In this process, the mysterious constant that appeared in the single crystal calculation vanished, leaving behind a clean result that depends only on the strength of the electric field and the material's dielectric properties. The resulting formula for the stress perpendicular to the electric field matched a famous result derived decades ago by the physicist Rudolf Peierls, who used a completely different method to study liquids. This agreement suggests that while the internal details of a crystal matter for the specific forces within it, the average behavior of a disordered solid aligns with established theories for liquids.

The study also explored what happens when the electric field points in an arbitrary direction rather than along a specific crystal axis. The researcher found that the stress in such a case is simply a combination of the stresses calculated for the three main axes, weighted by the direction of the field. This linear superposition confirmed that the complex behavior of the crystal can be understood by breaking it down into its simpler, symmetric components. Furthermore, the paper examined a simplified two-dimensional model of polarizable rods to illustrate how these forces behave in a signal, such as a pulse of light moving through a medium. In this model, the forces at the edges of the signal create an inward pull, which helps explain how momentum is carried by light in a liquid. The analysis showed that the ratio of momentum to energy in such a signal matches what is observed in real-world experiments, reinforcing the idea that the microscopic forces calculated in the study are physically real and measurable.

Ultimately, this work clarifies a long-standing ambiguity in physics by proving that the electromagnetic stress in a solid is not just a function of its bulk properties but is also influenced by the specific, microscopic arrangement of its atoms. While the standard macroscopic equations work well for many applications, they fail to capture the full picture of internal stress because they cannot account for the discrete nature of the atomic lattice. The research demonstrates that a complete understanding requires looking at the material at the atomic level, where the forces between individual dipoles create a pressure that is distinct from what macroscopic theory predicts. For amorphous materials, however, the randomness of the structure averages out these microscopic quirks, allowing the macroscopic description to hold true once again. This distinction provides a clear boundary for where the old rules apply and where the microscopic details become essential, offering a more precise foundation for understanding how light and matter interact at the most fundamental level.

Technical Summary: The Electromagnetic Stress Tensor in Cubic Crystals and Amorphous Solids

Problem Statement
Macroscopic electrodynamics successfully describes quasi-static phenomena and light propagation in condensed matter using constitutive coefficients like susceptibility. While the theory correctly reproduces bilinear energy density, it faces longstanding difficulties regarding the electromagnetic stress tensor (momentum flux). The standard phenomenological formula for the stress tensor involves the derivative of the dielectric tensor with respect to density ( $\partial \epsilon_r / \partial \rho$ ), a quantity that is often unknown and difficult to relate to microscopic averages. Furthermore, it is unclear whether macroscopic electrodynamics alone is sufficient to derive the stress tensor, particularly regarding constant terms that may arise from microscopic details. This paper addresses these limitations by calculating the stress tensor exactly for microscopic models of point-like atoms on a lattice.

Methodology
The author employs a microscopic model consisting of classical, point-like polarizable atoms arranged on an orthorhombic (and subsequently cubic) lattice. The atoms are modeled as interpenetrating spherically symmetric charge distributions with a radial scale $R$ much smaller than the lattice constants.

Microscopic Field Derivation: The electric field is derived using a Fourier series expansion of the charge density. This approach reproduces the discrete-dipole approximation and coincides with the Ewald-Kornfeld sum under specific convergence conditions. The derivation establishes a relationship between the microscopic polarizability and the macroscopic dielectric tensor components ( $\epsilon_{ii}$ ), recovering the Clausius-Mossotti relation for cubic crystals.
Direct Stress Tensor Calculation: The electromagnetic vacuum stress tensor, $\sigma^{\#}_{ij}$ , is computed directly from the microscopic electric field ( $E^{\#}$ ) and magnetic field ( $B^{\#}$ ) in the interstitial vacuum planes of the lattice. The calculation involves averaging the stress tensor over unit cell areas.
Phenomenological Comparison: The exact microscopic results are compared against a generalized phenomenological formula derived from energy balance arguments. This requires calculating the variation of the dielectric constant under uniaxial deformations of the lattice to determine the necessary derivatives ( $\partial \epsilon_{ii} / \partial \ell_i$ ).
Amorphous Modeling: To model amorphous solids, the author treats the system as a mosaic of randomly oriented grains of a cubic crystal. The stress tensor for the amorphous solid is deduced by averaging the cubic crystal stress tensor over all possible orientations of the electric field and the crystal axes.

Key Contributions and Results

Exact Microscopic Stress Tensor for Cubic Crystals: The paper derives explicit expressions for the stress tensor components in a cubic crystal.
- Perpendicular to the field ( $\sigma_{33}$ ): $\sigma^{cubic}_{33} = \frac{\epsilon_0}{2} E_1^2 [1 + (\epsilon_r - 1)^2 M]$ .
- Parallel to the field ( $\sigma_{11}$ ): $\sigma^{cubic}_{11} = -\frac{\epsilon_0}{2} E_1^2 [\epsilon_r^2 + (\epsilon_r - 1)^2 2M]$ .
- General Orientation: For an arbitrarily oriented field, the stress tensor includes a term dependent on a nontrivial constant $M \approx 0.1716$ , which arises from the summation of Fourier components over the lattice.
Validation of Phenomenological Formulas: The paper demonstrates that the exact microscopic results agree with the phenomenological formula (Eq. 20 and 21) only if the derivative of the dielectric tensor with respect to lattice deformation is calculated correctly using the microscopic model. This confirms the consistency between the microscopic and macroscopic approaches when the correct microscopic dependencies are included.
The Constant $M$ : A central finding is the presence of the constant $M$ in the stress tensor expressions. Because $M$ depends on the specific microscopic lattice summation and not just macroscopic parameters, the paper concludes that the electromagnetic stress tensor for cubic crystals cannot be derived from macroscopic electrodynamics alone.
Amorphous Solids: By averaging the cubic stress tensor over random orientations, the constant $M$ $M$ cancels out. The resulting stress tensor components for an amorphous solid are:
- Longitudinal (parallel to E): $\sigma^{\text{amorph}}_{\parallel} = -\frac{\epsilon_0}{2} E^2 [2\epsilon_r - 1 + \frac{3}{5}(\epsilon_r - 1)^2]$ .
- Transverse (perpendicular to E): $\sigma^{\text{amorph}}_{\perp} = \frac{\epsilon_0}{2} E^2 [1 - \frac{1}{5}(\epsilon_r - 1)^2]$ .
- The transverse component agrees with the result derived by Peierls for liquids using a different method.

Significance and Claims
The paper claims to resolve the ambiguity surrounding the electromagnetic stress tensor in condensed matter by providing an exact microscopic evaluation. Its primary significance lies in demonstrating that:

The stress tensor contains a nontrivial constant ( $M$ ) intrinsic to the lattice structure, which prevents a derivation solely from macroscopic electrodynamics for anisotropic (cubic) systems.
The phenomenological formula involving $\partial \epsilon / \partial \rho$ is valid but requires the specific microscopic dependence of the dielectric constant on deformation to yield the correct stress tensor.
In amorphous solids (modeled as random mosaics), the microscopic constant $M$ averages out, leading to stress tensor components that depend only on macroscopic parameters. The transverse component in this limit matches Peierls's result for liquids, providing a microscopic foundation for that earlier work.

The author notes that while dynamic phenomena (like light propagation) are generally beyond the scope of this static analysis, the derived stress tensor components are relevant for understanding the momentum density of signals in liquids, specifically regarding the ratio of momentum to energy observed in experiments.

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