Lorentz–Lorenz Local Field

Worked example: Finite ideal dielectric

Consider an ideal isotropic dilute dielectric with dimensionless density-polarizability combination η = 0.25 and a macroscopic electric field E = 1.

1. The relative permittivity is given by e_r = (1 + 2η) / (1 - η) = (1 + 0.5) / (1 - 0.25) = 1.5 / 0.75 = 2.0.
2. The refractive index is n = √e_r = √2.0 ≈ 1.4142.
3. The local field E_local = E / (1 - η) = 1 / 0.75 = 1.3333.
4. The macroscopic polarization scaled by the vacuum permittivity is P / ε₀ = (e_r - 1)E = (2.0 - 1) × 1 = 1.0.

Microscopic vs Macroscopic Field Response

Open Lorentz–Lorenz Local Field.

1. Select Polarizability density to 0.25 and Macroscopic field to 1. Observe the calculated Relative permittivity 2.0 and Local field 1.3333333333333333.
2. Change the Polarizability density to 0.5. Notice how the Local field increases to 2.0 and the Relative permittivity increases to 4.0. The macroscopic polarization P / ε₀ also increases to 3.0.
3. Change the Macroscopic field to 2. Observe that the Local field scales proportionally to 2.6666666666666665, and P / ε₀ scales to 2.0, but the Relative permittivity and Refractive index remain unchanged since this is a linear model.

Model Limitations

This linear model describes an ideal isotropic dilute nonmagnetic dielectric. η represents a dimensionless density-polarizability combination. As η approaches 1, the denominator (1 - η) goes to zero, causing the local field, permittivity, and refractive index to diverge.

This singularity at η = 1 indicates a model breakdown (a "polarization catastrophe"), and outputs become unavailable. This is not a verified real-material phase transition, nor does it serve as a universal law for dense resonant media or metals. The model also assumes negligible absorption and a relative permeability μ_r = 1.

Primary References

See the UT Austin physics lecture on the Lorentz local field for a detailed derivation of the local electric field in a dielectric and the Clausius-Mossotti (or Lorentz-Lorenz) relation linking microscopic polarizability to macroscopic permittivity.