Anisotropic Dielectrics and the Permittivity Tensor

The permittivity tensor is the most commercially deployed object in this chapter. Every liquid-crystal display pixel is a vat of rod-like molecules forming a tunable uniaxial dielectric: the nematic director (the local optic axis) reorients in an applied field because the molecules' $\Delta\varepsilon=\varepsilon_\parallel-\varepsilon_\perp$ makes alignment energetically favorable, and the reoriented tensor then rotates the polarization of transmitted light between crossed polarizers. A billion screens steer their tensors, pixel by pixel, sixty times a second. At the engineering frontier the game is designing $\varepsilon_{ij}$ rather than accepting what nature grows. Metamaterials assemble sub-wavelength cells whose effective tensors take values no crystal offers, including principal values of opposite sign (hyperbolic media); transformation optics runs our chapter backwards — choose the field pattern you want and solve for the spatially varying tensor that produces it. Even the humble circuit board participates: woven fiberglass laminate is measurably biaxial, and high-speed designers route differential pairs at an angle to the weave to dodge the resulting skew.

Picture a bound electron in a crystal as a mass caged by springs — stiff springs along one crystallographic direction, soft springs along another. A field applied exactly along a spring direction stretches only those springs, and the induced dipole points along the field. But push at $45^\circ$ and the charge cheats: it slides farther along the soft direction than the stiff one, so the dipole moment — and, summed over the lattice, the polarization $\mathbf{P}$ — leans away from $\mathbf{E}$ toward the compliant axis. The permittivity tensor is nothing more than the bookkeeping matrix of these direction-dependent stiffnesses.

Both microscopic ingredients of this cage are already on our shelf. Induced Dipoles and Atomic Polarizability warned that for anything less symmetric than a sphere, $\alpha$ is itself a tensor — a CO$_2$ molecule polarizes about twice as easily along its O=C=O axis as across it, a debt we pay in full below. And Local Field and the Lorentz Cavity showed that even perfectly isotropic molecules acquire an anisotropic response from an anisotropic lattice: the near-field sum that vanishes identically on a cubic lattice revives at the 14% level under a mere 5% tetragonal stretch, turning the local-field correction — and hence the susceptibility — into a direction-dependent object. Molecular shape and lattice geometry are the two roads into the tensor; most real crystals travel both. The theme of this article is that the tensor is far tamer than nine arbitrary numbers. Energy conservation makes it symmetric; symmetry makes it diagonalizable; and the crystal's own point symmetry then squeezes the three surviving eigenvalues, sometimes all the way back to one.

For an isotropic linear dielectric we wrote $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$, one scalar multiplying the total macroscopic field. To let the medium respond differently in different directions, we keep linearity — still valid for the small-field Taylor reasons of Linear Dielectrics, Susceptibility and Permittivity — but promote the single coefficient to a matrix,

where: $P_i$ — components of the polarization density (C/m²); $E_j$ — components of the total macroscopic field, the repeated index $j$ summed over $x,y,z$; $\chi_{ij}$ — the electric susceptibility tensor, nine dimensionless numbers. Read it as a machine: feed in a field along $\hat{\mathbf{y}}$ and the column $(\chi_{xy},\chi_{yy},\chi_{zy})$ tells you the polarization it produces, off-axis components included. Diagonal $\chi_{ij}=\chi_e\delta_{ij}$ recovers the isotropic case exactly. The displacement keeps its universal definition $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ — that equation, from The Electric Displacement D, is bookkeeping and holds in any medium whatsoever. Only the constitutive link closing the equations changes:

where: $\varepsilon_{ij}$ — the permittivity tensor (F/m); $\delta_{ij}$ — the Kronecker delta carrying the vacuum part. Because $\chi_{ij}$ need not be proportional to $\delta_{ij}$, $\mathbf{D}$ and $\mathbf{E}$ point in different directions unless $\mathbf{E}$ lies along one of the special directions we construct below. Gauss's law in dielectrics is untouched: $\nabla\cdot\mathbf{D}=\rho_f$ cares only about $\mathbf{D}$'s divergence, not about which way it leans relative to $\mathbf{E}$. Where do the nine $\chi_{ij}$ come from? Sum the microscopic responses, as in the isotropic chapter but keeping directions. If each molecule carries a polarizability tensor $\alpha_{ij}$ (so $p_i=\alpha_{ij}E^{\mathrm{loc}}_j$) and sits in a lattice whose local-field correction is itself directional, $\chi_{ij}$ inherits anisotropy from both factors. For CO$_2$: $\alpha_\parallel\approx4.8\times10^{-40}\ \mathrm{C\,m^2/V}$ along the molecular axis versus $\alpha_\perp\approx2.5\times10^{-40}$ across it (polarizability volumes $4.3$ and $2.2\ \text{Å}^3$) — two-to-one anisotropy in a single molecule. In the gas this washes out: tumbling molecules present the average $\bar{\alpha}=(\alpha_\parallel+2\alpha_\perp)/3$ and the bulk stays isotropic, as we verify numerically in the worked examples. Freeze the same molecules into an ordered lattice and the anisotropy survives to the macroscale — that is the crystal.

Nine components suggest nine material constants. Thermodynamics immediately strikes three: for any medium that stores field energy without loss, $\varepsilon_{ij}=\varepsilon_{ji}$. This is not a convenience assumption — its violation would let you build a perpetual-motion machine out of a capacitor, and the derivation shows exactly how.

With symmetry in hand, the energy density itself comes along for free. Since $\mathrm{d}u=\varepsilon_{ij}E_i\,\mathrm{d}E_j$ is exact precisely when the tensor is symmetric, it integrates to

where: $u$ — electrostatic energy density (J/m³), a positive-definite quadratic form for any passive dielectric. The familiar $\tfrac12\mathbf{E}\cdot\mathbf{D}$ of the isotropic chapter survives verbatim — with the scalar product now doing real work, because the two vectors are no longer parallel. The which-energy honesty of Energy Stored in Dielectric Systems also carries over unchanged: this $u$ is the free-charge ledger, springs included.

A real symmetric matrix is the friendliest object in linear algebra: it can always be diagonalized by a pure rotation. Physically, every crystal — however low its symmetry — hides three mutually perpendicular directions along which the response is parallel to the push.

One geometric consequence is worth carrying in your head: since $D_i=\partial u/\partial E_i$, $\mathbf{D}$ points along the normal to the constant-$u$ ellipsoid at the tip of $\mathbf{E}$ — and an ellipse's normal is parallel to its radius only on the axes. That is the picture behind every walk-off angle we compute:

Six independent components is the worst case — triclinic, no symmetry beyond the lattice itself. Every symmetry operation the crystal possesses is a constraint, by Neumann's principle: a macroscopic property tensor must be invariant under every point-symmetry operation of its crystal, $R\,\boldsymbol{\varepsilon}\,R^{\mathsf{T}}=\boldsymbol{\varepsilon}$ for each symmetry rotation or reflection $R$ — the property cannot be choosier than the crystal that hosts it. The payoff is a complete classification, and one genuinely surprising theorem.

Anchor numbers, at 589 nm where the anisotropic tensor is best tabulated: calcite is negative uniaxial with $\varepsilon_o=n_o^2\,\varepsilon_0=2.75\,\varepsilon_0$, $\varepsilon_e=2.21\,\varepsilon_0$ ($n_o=1.658$, $n_e=1.486$); quartz is weakly positive ($n_o=1.544$, $n_e=1.553$); rutile dramatically so ($n_o=2.616$, $n_e=2.903$). One honesty note from Linear Dielectrics: $\varepsilon=\varepsilon_0n^2$ is the optical-frequency tensor. At DC the ionic lattice adds its own stretch response — calcite's static values sit near $8.5\,\varepsilon_0$ and $8.0\,\varepsilon_0$ — and the full frequency story is dispersion, deferred to the waves chapter.

Example 1 — calcite: how far does D lean? A slab of calcite carries a uniform field $E=2.0\times10^5$ V/m at $45.0°$ to the optic axis (optical-frequency tensor: $\varepsilon_o=2.749\,\varepsilon_0$, $\varepsilon_e=2.208\,\varepsilon_0$). Find $\mathbf{D}$, the walk-off angle, and the stored energy density.

Step 1. Set up in the principal frame, optic axis along $z$, field in the $xz$-plane: $E_x=E\sin45°=1.414\times10^5$ V/m, $E_z=E\cos45°=1.414\times10^5$ V/m.

Step 2. Apply the diagonal tensor component by component: $D_x=\varepsilon_oE_x=2.749\times(8.854\times10^{-12})\times1.414\times10^5=3.44\times10^{-6}$ C/m², and $D_z=\varepsilon_eE_z=2.77\times10^{-6}$ C/m². The magnitude is $|\mathbf{D}|=4.42\times10^{-6}$ C/m².

Step 3. Direction of $\mathbf{D}$: $\tan\theta_D=D_x/D_z=(\varepsilon_o/\varepsilon_e)\tan\theta_E=1.245$, so $\theta_D=51.2°$ — the displacement leans toward the easy plane. The walk-off angle is $\rho=\theta_D-\theta_E=$ 6.2°. (Sweeping $\theta_E$ shows the lean maximizes at $6.26°$ for $\theta_E=41.9°$, the analytic maximum $\arctan[(r-1)/2\sqrt{r}\,]$ with $r=\varepsilon_o/\varepsilon_e$.)

Step 4. Energy density: $u=\tfrac12\mathbf{E}\cdot\mathbf{D}=\tfrac12(E_xD_x+E_zD_z)=$ 0.439 J/m³. Cross-checks: the quadratic form $\tfrac12(\varepsilon_oE_x^2+\varepsilon_eE_z^2)$ and the projection $\tfrac12|\mathbf{E}||\mathbf{D}|\cos6.2°$ give the same $0.439$ J/m³ — the $\cos\rho$ is the anisotropy tax on the naive $\tfrac12|E||D|$.

Example 2 — CO2, from one molecule to a gas. The CO$_2$ polarizability tensor has $\alpha_\parallel=4.80\times10^{-40}$ C·m²/V along the molecular axis and $\alpha_\perp=2.46\times10^{-40}$ C·m²/V across it. (a) A field $E=3.0\times10^6$ V/m is applied at $45°$ to a molecule's axis — where does $\mathbf{p}$ point? (b) Predict the dielectric constant of CO$_2$ gas at 20 °C, 1 atm, and compare with the measured $\kappa=1.000922$.

Step 1. Single molecule, principal frame along the O=C=O axis: $p_\parallel=\alpha_\parallel E\cos45°=1.02\times10^{-33}$ C·m, $p_\perp=\alpha_\perp E\sin45°=5.2\times10^{-34}$ C·m. Direction: $\tan\theta_p=(\alpha_\perp/\alpha_\parallel)\tan45°=0.513$, so $\mathbf{p}$ sits $27.2°$ from the axis — tilted $17.8°$ away from $\mathbf{E}$, toward the easy axis. A single molecule is a miniature calcite.

Step 2. In the gas the molecules tumble, and the orientation average of the tensor is the scalar $\bar{\alpha}=(\alpha_\parallel+2\alpha_\perp)/3=3.24\times10^{-40}$ C·m²/V — one soft axis, two stiff ones, weighted equally. The macroscopic response is isotropic even though no single molecule is.

Step 3. Number density at 20 °C, 1 atm: $N=p/k_BT=101325/(1.381\times10^{-23}\times293.15)=2.50\times10^{25}$ m⁻³. The gas is dilute enough ($N\bar{\alpha}/3\varepsilon_0\sim10^{-4}$) that the Clausius–Mossotti denominator is negligible and $\chi_e\approx N\bar{\alpha}/\varepsilon_0=9.16\times10^{-4}$.

Step 4. Prediction: $\kappa=1+\chi_e=$ 1.000916, against the measured $1.000922$ — agreement to better than 1% of the susceptibility. The anisotropy has not vanished, only hidden: light scattered from the gas is measurably depolarized because individual molecules still respond off-axis, which is exactly how $\alpha_\parallel-\alpha_\perp$ is measured. Freeze the gas into an oriented crystal and the $17.8°$ tilt of Step 1 would survive to the macroscale.

References. Griffiths, Introduction to Electrodynamics, 4th ed., §4.4 (linear dielectrics; anisotropic remark) · Jackson, Classical Electrodynamics, 3rd ed., §4.5 and §7.10 (crystal optics) · Zangwill, Modern Electrodynamics, §6.5 (dielectric tensor) · Born & Wolf, Principles of Optics, Ch. 15 (crystal optics, index ellipsoid) · Landau & Lifshitz, Electrodynamics of Continuous Media, §13, §97-98 · Nye, Physical Properties of Crystals, Ch. 1-3 (Neumann's principle, tensor census)

Provenance. Tensor constitutive relation, symmetry from energy, diagonalization, index ellipsoid, and the uniaxial Fresnel equation follow Jackson §4.5 & §7.10, Born & Wolf Ch. 15, and Landau-Lifshitz ECM §13,97. Verifier should recompute: (1) symmetry of $\varepsilon_{ij}$ from exactness of $\mathrm{d}u=E_i\,\mathrm{d}D_i$; (2) the uniaxial index formula $1/n_e^2(\theta)=\cos^2\theta/n_o^2+\sin^2\theta/n_e^2$ from the plane-wave wave equation; (3) calcite values $n_o\approx1.658$, $n_e\approx1.486$ at 589 nm (standard handbook values); (4) the component counts (cubic 1, uniaxial 2, biaxial 3) against Nye's crystal-symmetry tables.

Builds on: Linear Dielectrics, Susceptibility and Permittivity · The Electric Displacement D · Polarization Density P · Induced Dipoles and Atomic Polarizability · Gauss's Law in Dielectrics · Local Field and the Lorentz Cavity Leads to: Boundary Conditions for D and E · Magnetostatics (the chapter that opens next) · Birefringence and Crystal Optics · Electro-optic Effect (Pockels and Kerr)