Lorentz oscillator susceptibility

Let a particle of charge $q$ and mass $m$ move by a small displacement $x$ from equilibrium. A harmonic restoring force supplies a natural angular frequency $\omega_0$, while a positive damping rate $\gamma$ removes mechanical energy. The applied macroscopic field is taken as the driving field, an approximation that absorbs local-field effects into fitted parameters. The displacement must remain small enough for both the restoring law and the polarization response to be linear.

Insert the convention $e^{-i\omega t}$ into the mechanical equation. Acceleration contributes a negative squared frequency, while damping contributes a negative imaginary term in the denominator. Multiply the displacement by its charge to form a dipole, then multiply by the number density $N$ of identical oscillators. The charge appears squared in the susceptibility, so an electron's negative charge does not make an ordinary low-frequency dielectric susceptibility negative. It determines the displacement direction before the dipole is formed.

Write the real detuning as the difference of squared natural and driving frequencies. Multiplying numerator and denominator by the complex conjugate separates the real and imaginary response. Below resonance the real part is positive; above resonance it becomes negative for this isolated oscillator contribution. Exactly at the undamped natural frequency the real contribution vanishes, while the imaginary contribution is positive. That statement concerns the susceptibility, not necessarily the full permittivity with background contributions included.

The polarization current amplitude is $-i\omega P$. Averaging its product with the real electric field gives a positive material heating rate proportional to the imaginary susceptibility. This is the physical reason the sign of the damping term matters. With zero damping, driving exactly at resonance does not produce a finite stationary response: energy accumulates and the steady harmonic ansatz fails there. Taking a small positive damping before a resonant limit avoids hiding that singular behavior.

Far below resonance, the charge follows the field almost quasistatically and the susceptibility approaches a positive constant. Far above resonance, inertia dominates and the response falls as the inverse squared frequency with a negative real sign. A finite damping smooths the transition and gives a finite linewidth. The driving frequency of maximum displacement is slightly shifted from the natural frequency when damping is appreciable; different observables need not peak at exactly the same frequency.

A real dielectric can require a sum of resonances and an approximately constant background over the measured band. The model parameters then summarize a chosen frequency range rather than revealing one literal spring per atom. The oscillator lab illustrates how a prepared displacement loses amplitude after release. Its initial velocity is zero and it has no continuing drive. Consequently its decay rate can support intuition about linewidth, but its displayed motion cannot be read as a driven susceptibility spectrum.