Dielectric Susceptibility and Permittivity

We take one result from electromagnetism and give it a frequency. The companion article aq-em-linear-dielectrics-susceptibility-and-permittivity establishes, for static fields, that a linear dielectric polarizes in proportion to the field it sits in: $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ , that the displacement field then collapses to $\mathbf{D}=\varepsilon_0(1+\chi_e)\mathbf{E}$ , and that the dimensionless ratio $\varepsilon_r=1+\chi_e$ is the dielectric constant. That is the electrostatics, and it is owned there. Optics does one thing to it: it lets the field oscillate. Drive the same bound electrons at $10^{15}$ hertz and they can no longer follow the field instantaneously — the response lags, dissipates, and depends sharply on how close the drive frequency sits to a resonance. The single real constant $\chi_e$ becomes a complex, frequency-dependent function $\chi(\omega)=\chi'(\omega)+i\chi''(\omega)$ , and that function is the whole optical personality of a material. Everything measurable about how a transparent medium treats light is buried in $\chi(\omega)$ . Its real part $\chi'$ sets the refractive index through $n=\sqrt{1+\chi'}$ and hence the dispersion that fans a prism; its imaginary part $\chi''$ sets the absorption that colours a stained-glass window and defines every spectroscopic line. The two are not independent — causality, through the Kramers–Kronig relations, forces a material that bends light to absorb it somewhere. Get $\chi(\omega)$ and you have the index, the absorption spectrum, the group delay, and the seed of every nonlinear effect. This article is about that one response function: what it is, how it repackages into the permittivity a designer actually looks up, and why its frequency dependence is the entire subject of the chapters that follow.

Key takeaways

The electric susceptibility $\chi$ is the dimensionless linear-response coefficient relating induced polarization to field, $\mathbf{P}=\varepsilon_0\chi\mathbf{E}$ ; the permittivity is just its repackaging, $\varepsilon=\varepsilon_0(1+\chi)$ , $\varepsilon_r=1+\chi$ .
The optical payoff is one line: for a non-magnetic medium $n=\sqrt{\varepsilon_r}=\sqrt{1+\chi}$ — the index is the square root of "one plus the susceptibility".
Electromagnetism owns the statics ( $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ , $\varepsilon_r=1+\chi_e$ ); optics takes $\chi\to\chi(\omega)$ , complex and frequency-dependent, because bound electrons cannot follow an optical field in phase.
$\chi'(\omega)$ (real part) governs the index and dispersion; $\chi''(\omega)$ (imaginary part) governs absorption; causality ties them by the Kramers–Kronig relations.
The low-frequency dielectric constant is not $n^2$ : water has $\varepsilon_r\approx80$ statically but $n^2\approx1.77$ optically, because slow orientational polarization freezes out at optical frequencies.
Key idea: one dimensionless response function $\chi(\omega)$ converts a statement about bound charges into all of a material's linear optics — through $\varepsilon=\varepsilon_0(1+\chi)$ and $n=\sqrt{1+\chi}$ , its real part is the index and its imaginary part is the absorption.

The core insight

Electromagnetism gives the static linear-dielectric law $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}\Rightarrow\varepsilon_r=1+\chi_e$ ; optics promotes the constant to a complex frequency-dependent function $\chi(\omega)$ whose real part fixes the refractive index $n=\sqrt{1+\chi'}$ and whose imaginary part fixes the absorption. One response function carries all the linear optics of the medium.

Where this lives today

The susceptibility is the design parameter of photonics. Choosing glasses for a camera lens, a fibre, a waveguide, or an anti-reflection stack is, at bottom, choosing $\varepsilon_r(\omega)=1+\chi(\omega)$ across the transparency window; the Sellmeier equation used everywhere is a multi-resonance fit to $\chi'(\omega)$ . Absorption spectroscopy is the direct measurement of $\chi''(\omega)$ : each molecular fingerprint peak is a resonance in the imaginary susceptibility, and its width reports the damping. And the entire field of nonlinear optics — frequency doublers, optical parametric oscillators, the Kerr lensing that mode-locks a Ti:sapphire laser — lives in the higher terms $\chi^{(2)},\chi^{(3)},\dots$ that appear the moment the field is strong enough that the linear $\chi$ treated here is only the first term of a series. Every one of those technologies begins by writing down a susceptibility.

The physical picture

Model each atom as a negative electron cloud tethered to its nucleus by a spring. Switch on a field and the cloud shifts, making a dipole; add the dipoles per unit volume and you have the polarization $\mathbf{P}$ . The susceptibility $\chi$ is the bookkeeping of stiffness and density — a large $\chi$ means the charge displaces easily, or there is a lot of it. The permittivity is then a labour-saving fiction: rather than track the bound charges explicitly, we fold them into $\varepsilon$ and pretend the medium is "vacuum with a bigger constant", so Maxwell's equations keep their vacuum form under $\varepsilon_0\to\varepsilon$ . The first figure shows that repackaging.

Figure

The response chain: the field E drives bound charge into a polarization P = ε₀χE, and D = ε₀E + P = ε₀(1+χ)E hides the bound charge inside a single constant ε

Frequency is the whole story because springs have resonances. Far below a resonance the electrons track the field in phase, $\chi$ is real and slowly varying, and the medium is transparent and normally dispersive. Approach a resonance and the electrons lag and dissipate: $\chi$ develops a large imaginary part — an absorption line — and its real part swings violently, the anomalous-dispersion wiggle. A complex $\chi=\chi'+i\chi''$ is not a trick of notation but a statement about timing: with the convention $\mathbf{E}\propto e^{-i\omega t}$ , a positive $\chi''$ means the polarization lags the field, so the field does net work on the charges and energy is absorbed. The second figure is the single most important picture in the chapter — the frequency-dependent susceptibility itself.

Figure

The optical susceptibility χ(ω) near one resonance: the real part χ′ (dispersion, an anomalous wiggle across ω₀) and the imaginary part χ″ (absorption, a peak at ω₀), with n = √(1+χ′) giving glass χ = 1.25 → n = 1.50 and air χ = 5.85×10⁻⁴ → n = 1.0002926

The core equations

The defining relation is the linear, isotropic, local response: the polarization at a point is proportional to the field at that same point, with a dimensionless proportionality constant. Below it is a definition of $\chi$ , not a law of nature — real materials obey it only in the weak-field, single-frequency limit.

\mathbf{P}=\varepsilon_0\,\chi\,\mathbf{E} \qquad (1)

where: $\mathbf{P}$ — polarization density (dipole moment per unit volume, $\mathrm{C/m^2}$ ); $\varepsilon_0$ — vacuum permittivity, $8.854\times10^{-12}\ \mathrm{F/m}$ ; $\chi$ — electric susceptibility (dimensionless); $\mathbf{E}$ — macroscopic electric field. This is exactly the static relation recalled by name from aq-em-linear-dielectrics-susceptibility-and-permittivity; the optics enters only when $\chi$ is allowed to depend on $\omega$ . Feeding this into the definition of the displacement field folds the polarization into a single material constant. The permittivity $\varepsilon$ is therefore not independent physics — it is a compressed way of writing "vacuum plus a susceptibility".

\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}=\varepsilon_0(1+\chi)\mathbf{E}=\varepsilon\,\mathbf{E},\qquad \varepsilon_r=\frac{\varepsilon}{\varepsilon_0}=1+\chi \qquad (2)

where: $\mathbf{D}$ — displacement field ( $\mathrm{C/m^2}$ ); $\varepsilon$ — permittivity ( $\mathrm{F/m}$ ); $\varepsilon_r$ — relative permittivity (dielectric constant), dimensionless and $\ge 1$ for ordinary dielectrics. The number a glass catalogue lists is $\varepsilon_r$ , and equation (2) says it is nothing but $1+\chi$ . The one optical consequence that justifies the whole exercise is the refractive index. For a medium that is non-magnetic at optical frequencies — essentially all of them, since $\mu_r\to1$ — the index is fixed entirely by the susceptibility.

n=\sqrt{\varepsilon_r\,\mu_r}\;\xrightarrow{\ \mu_r=1\ }\;\sqrt{1+\chi} \qquad (3)

where: $n$ — refractive index; $\mu_r$ — relative permeability ( $\approx1$ optically). Derivation A obtains this from the wave equation; it is the bridge from "bound-charge response" to "how much light slows". At optical frequencies the response is not instantaneous, so the susceptibility carries a phase. With the physics convention $\mathbf{E}\propto e^{-i\omega t}$ , splitting $\chi$ into real and imaginary parts separates dispersion from dissipation.

\chi(\omega)=\chi'(\omega)+i\,\chi''(\omega) \qquad (4)

where: $\chi'$ — in-phase (dispersive) response, sets $n$ ; $\chi''\ge0$ — quadrature (absorptive) response, sets attenuation; $\omega$ — angular frequency of the driving field. A positive $\chi''$ is a polarization that lags the field, which is exactly a medium doing negative work on the wave. Because a physical medium cannot respond before it is driven, $\chi(\omega)$ is the Fourier transform of a causal time-domain kernel, and that single fact links its real and imaginary parts. The Kramers–Kronig relation below is stated for reference; the causality argument and its consequences are developed in the companion kramers-kronig-relations.

\chi'(\omega)=\frac{2}{\pi}\,\mathcal{P}\!\int_0^{\infty}\!\frac{\omega'\,\chi''(\omega')}{\omega'^2-\omega^2}\,\mathrm{d}\omega' \qquad (5)

where: $\mathcal{P}$ — Cauchy principal value. The physical reading is stark: the entire dispersion curve $\chi'(\omega)$ is determined by the absorption spectrum $\chi''(\omega)$ at all frequencies, and vice versa. The explicit resonant shape of $\chi(\omega)$ — the Lorentzian drawn above — is the content of the Lorentz oscillator model, recalled by name from aq-optics-lorentz-oscillator-model and not re-derived here.

Derivation APermittivity and refractive index from the susceptibility

The goal is to show that the two "extra" field quantities of dielectric optics — the permittivity $\varepsilon$ and the refractive index $n$ — are not new physics but repackagings of the single susceptibility $\chi$ , and to trace the chain from the constitutive relation all the way to the speed of light in the medium. We start from the definition of $\mathbf{D}$ , insert the linear response, then feed the resulting $\mathbf{D}=\varepsilon\mathbf{E}$ into Maxwell's source-free wave equation to read off the phase velocity and hence $n$ . This is where the optics of a material collapses onto one number.

Step 1. Write the always-valid definition of the displacement field, $\mathbf{D}\equiv\varepsilon_0\mathbf{E}+\mathbf{P}$ , and substitute the linear constitutive relation $\mathbf{P}=\varepsilon_0\chi\mathbf{E}$ to get $\mathbf{D}=\varepsilon_0(1+\chi)\mathbf{E}\equiv\varepsilon\mathbf{E}$ with $\varepsilon\equiv\varepsilon_0(1+\chi)$ . Why: $\mathbf{D}$ is defined for any medium, but only a linear one lets us factor the common field and name the coefficient $\varepsilon$ — the entire dielectric response has been compressed into one constant.

Step 2. In a homogeneous, source-free, non-magnetic dielectric the macroscopic curl equations (recalled by name from aq-optics-maxwell-equations-dielectric-media) combine into the wave equation $\nabla^2\mathbf{E}=\mu_0\varepsilon\,\partial_t^2\mathbf{E}$ , a standard wave equation with speed $v=1/\sqrt{\mu_0\varepsilon}$ . Why: the constitutive relation is what turns the two first-order Maxwell equations into a single second-order wave equation whose coefficient carries the material physics.

Step 3. Substitute $\varepsilon=\varepsilon_0(1+\chi)$ and use $c=1/\sqrt{\mu_0\varepsilon_0}$ : $v=1/\sqrt{\mu_0\varepsilon_0(1+\chi)}=c/\sqrt{1+\chi}$ . Why: the vacuum speed factors out cleanly because only $\varepsilon$ , not $\mu_0$ , changed — the medium slows light purely through its polarizability.

Step 4. Read off the index $n=c/v=\sqrt{1+\chi}=\sqrt{\varepsilon_r}$ ; it reduces to $n=1$ for $\chi=0$ (vacuum) and exceeds one for any ordinary dielectric. Why: this is the owned result — a statement about how easily bound charge moves has become a statement about how much light slows, valid frequency-by-frequency once $\chi\to\chi(\omega)$ . *Verified symbolically ( $\varepsilon_r=1+\chi$ ; $n=\sqrt{1+\chi}$ ; $n\to1$ as $\chi\to0$ ) and numerically (glass $\varepsilon_r=2.25\Rightarrow\chi=1.25\Rightarrow n=1.50$ ) in verify_aq-optics-dielectric-susceptibility-permittivity.py.*

Go deeper · Derivation B · 1 minComplex susceptibility, complex index, and absorption

Derivation BComplex susceptibility, complex index, and absorption

The aim here is to show why a complex susceptibility forces a complex refractive index, and to extract from its imaginary part the intensity absorption coefficient that a spectrometer actually reads. We take the weak-response (dilute) limit where $|\chi|\ll1$ , expand $n=\sqrt{1+\chi}$ to first order, and follow a plane wave carrying the complex index to see the real part become phase and the imaginary part become decay. The result is that $\chi''$ alone controls how a beam dims.

Step 1. Write the complex index $\tilde n=\sqrt{1+\chi}=n+i\kappa$ with $\chi=\chi'+i\chi''$ ; in the weak-response limit $|\chi|\ll1$ expand $\tilde n\approx1+\tfrac12(\chi'+i\chi'')$ , so $n\approx1+\tfrac12\chi'$ and $\kappa\approx\tfrac12\chi''$ . Why: gases and dilute media have $\chi$ of order $10^{-4}$ , so the square root linearizes and the real and imaginary parts of the index inherit those of the susceptibility directly.

Step 2. Propagate a plane wave $E\propto e^{i(\tilde n\omega z/c-\omega t)}=e^{i(n\omega z/c-\omega t)}\,e^{-\kappa\omega z/c}$ ; the real part of $\tilde n$ advances the phase, the imaginary part attenuates the amplitude. Why: putting a complex index into the propagation factor splits it automatically into an oscillation and an exponential envelope — dispersion and absorption fall out of the same object.

Step 3. Form the intensity $I=|E|^2\propto e^{-\alpha z}$ with absorption coefficient $\alpha=2\kappa\omega/c=4\pi\kappa/\lambda_0$ , and use $\kappa\approx\tfrac12\chi''$ to get $\alpha\approx\omega\chi''/c$ . Why: intensity is amplitude squared, doubling the decay exponent; the Beer–Lambert law is thus a direct read-out of $\chi''$ , which is why absorption spectroscopy maps the imaginary susceptibility. *Verified: the weak-field expansion $n\approx1+\tfrac12\chi'$ , $\alpha=4\pi\kappa/\lambda_0$ , and the air case $\chi=5.85\times10^{-4}\Rightarrow n=1.0002926$ in verify_aq-optics-dielectric-susceptibility-permittivity.py.*

Limiting cases

Case	Result	Meaning
Vacuum	$\chi=0,\ \varepsilon=\varepsilon_0,\ n=1$	no bound charge to polarize; the field propagates at $c$
Dilute gas	$\chi=N\alpha_p/\varepsilon_0\ll1,\ n\approx1+\tfrac12\chi$	$n-1$ linear in number density $N$ (hence pressure); air $\chi\approx5.85\times10^{-4}$ , $n-1\approx2.9\times10^{-4}$
Dense / polar medium	$\dfrac{\varepsilon_r-1}{\varepsilon_r+2}=\dfrac{N\alpha_p}{3\varepsilon_0}$	Clausius–Mossotti: the local field differs from the macroscopic field, so $\chi\ne N\alpha_p/\varepsilon_0$ (local-field owner recalled by name)
Anisotropic crystal	$P_i=\varepsilon_0\chi_{ij}E_j$	$\chi_{ij}$ a symmetric tensor; $\mathbf{P}\nparallel\mathbf{E}$ , the origin of birefringence
Strong field	$P_i=\varepsilon_0(\chi^{(1)}_{ij}E_j+\chi^{(2)}_{ijk}E_jE_k+\cdots)$	higher orders generate harmonics and the Kerr effect — all of nonlinear optics

Common trap“The dielectric constant you look up is n².”

It is not — not unless you look it up at the right frequency. The susceptibility is strongly frequency-dependent, so the static dielectric constant and the optical $n^2$ are different numbers whenever slow polarization mechanisms exist. Water is the classic trap: its static $\varepsilon_r\approx80$ (the huge value that makes it a good solvent) would predict $n=\sqrt{80}\approx8.9$ , whereas water's optical index is $n\approx1.33$ , i.e. $n^2\approx1.77$ . The difference is orientational polarization — the permanent dipoles of the polar molecules reorient freely in a slowly varying field but cannot keep up with a $10^{15}$ -hertz optical field, so that enormous contribution to $\chi$ simply switches off and only the fast electronic response survives. Always quote $\chi$ , $\varepsilon_r$ , or $n$ at the frequency you mean.

Worked examples

1 · From dielectric constant to index (confidence builder)

An optical glass has relative permittivity $\varepsilon_r=2.25$ at the sodium-D frequency. Find its susceptibility and refractive index, and the wave's phase velocity.

Step 1. Susceptibility directly from equation (2): $\chi=\varepsilon_r-1=2.25-1=\mathbf{1.25}$ . The bound-charge response contributes $1.25$ on top of the vacuum's $1$ .

Step 2. Refractive index from equation (3): $n=\sqrt{\varepsilon_r}=\sqrt{2.25}=\mathbf{1.50}$ — the value for ordinary crown glass, and the reason a glass block slows light to two-thirds of $c$ .

Step 3. Phase velocity: $v=c/n=2.998\times10^8/1.50=\mathbf{2.00\times10^8\ m/s}$ , and the in-medium wavelength shrinks by the same factor, $\lambda=\lambda_0/n$ . χ = 1.25, n = 1.50, v = 2.00×10⁸ m/s.

2 · The susceptibility of air (dilute-gas limit)

Air at standard conditions has refractive index $n=1.0002926$ for visible light. Find its electric susceptibility, and confirm the dilute-gas approximation $n\approx1+\tfrac12\chi$ .

Step 1. Exactly, $\chi=n^2-1=(1.0002926)^2-1$ . Expanding, $n^2-1=2(n-1)+(n-1)^2=2(2.926\times10^{-4})+(2.926\times10^{-4})^2$ ; the squared term is $\sim10^{-7}$ , negligible, giving $\chi=\mathbf{5.85\times10^{-4}}$ .

Step 2. Check the dilute approximation: $\tfrac12\chi=2.93\times10^{-4}$ , so $1+\tfrac12\chi=1.000293$ , matching $n=1.0002926$ to five figures. Why it works: with $\chi\sim10^{-4}$ the square root $\sqrt{1+\chi}$ is linear to one part in $10^{7}$ , so for gases the index offset $n-1$ is simply half the susceptibility.

Step 3. The physical consequence: because $\chi=N\alpha_p/\varepsilon_0$ for a dilute gas, $n-1\propto N\propto$ pressure. Halve the air pressure and you halve $n-1$ — the basis of the pressure corrections applied in precision interferometry and astrometry. χ_air = 5.85×10⁻⁴, and n − 1 = χ/2 to five figures.

Physics of the portrait

The susceptibility is the quiet workhorse of optics: every refractive index, every absorption line, every nonlinear coefficient is a moment of this one response function. Its story is the nineteenth-century effort to connect a bulk "dielectric constant" to what individual molecules do.

Origins · 1 minelectric susceptibility and how this was discovered

electric susceptibility

linear optical response function · electromagnetic optics of media

The electric susceptibility began as an empirical bulk constant — Faraday's "specific inductive capacity" of 1837, the factor by which a dielectric increased a capacitor's charge. The conceptual leap was to derive it from molecules. Ottaviano-Fabrizio Mossotti in 1850 and, independently and more completely, Rudolf Clausius in 1879 related the bulk permittivity to the polarizability of individual molecules and their number density, arriving at the Clausius–Mossotti relation $\tfrac{\varepsilon_r-1}{\varepsilon_r+2}=\tfrac{N\alpha_p}{3\varepsilon_0}$ — the first bridge from micro to macro, and the reason a susceptibility is not simply $N\alpha_p/\varepsilon_0$ once the local field a molecule feels differs from the average field. Hendrik Lorentz and Ludvig Lorenz cast the same content in optical form (the Lorentz–Lorenz relation for $n^2$ ) around 1880. The final piece was Peter Debye's 1912 treatment of polar molecules, whose permanent dipoles reorient with the field: this orientational susceptibility is enormous at low frequency but freezes out at optical frequencies, and it is exactly why water's static $\varepsilon_r\approx80$ collapses to an optical $n^2\approx1.77$ . The modern reading — $\chi(\omega)$ as a causal, complex, frequency-dependent response function tied together by Kramers–Kronig — is the culmination of that lineage.

History

1837

— Faraday introduces specific inductive capacity, the first "dielectric constant"

1850

— Mossotti relates bulk permittivity to molecular polarizability

1879

— Clausius completes the Clausius–Mossotti relation

1880

— Lorentz and Lorenz give the optical form (Lorentz–Lorenz relation for n²)

1912

— Debye treats polar molecules, explaining the frequency collapse of ε_r

Going further

Texts. Hecht, Optics, 5th ed., §3.5 (the electromagnetic approach; dispersion) · Jackson, Classical Electrodynamics, 3rd ed., §4.3 and §7.5 (linear response, dispersion) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §5.2 (monochromatic light in linear media), Ch. 21 (nonlinear susceptibility) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1–2 (dielectric response) · Demtröder, Laser Spectroscopy, Vol. 1, §3.1 (absorption and dispersion).

Provenance. The static linear-dielectric relations $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ , $\varepsilon_r=1+\chi_e$ are recalled by name from the electromagnetism owner aq-em-linear-dielectrics-susceptibility-and-permittivity and verified there; this article establishes only their optical (frequency-dependent) extension. A verifier should confirm: (1) $\varepsilon_r=1+\chi$ and $n=\sqrt{1+\chi}$ by substitution into $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ and the wave equation (Hecht §3.5; Jackson §4.3); (2) the weak-field expansion $n\approx1+\tfrac12\chi'$ , $\alpha=4\pi\kappa/\lambda_0$ (Loudon Ch. 1–2). Constants $\varepsilon_0=8.854\times10^{-12}\,\mathrm{F/m}$ , $e$ , $m_e$ are CODATA values; glass $n\approx1.5$ , air $n-1\approx2.9\times10^{-4}$ , water static $\varepsilon_r\approx80$ / optical $n\approx1.33$ are standard textbook figures. The Lorentzian shape of $\chi(\omega)$ is derived in aq-optics-lorentz-oscillator-model (forward reference, not re-derived here).

Prerequisites. Electric Polarization and the Displacement Field · Maxwell's Equations in Dielectric Media. Leads to. The Refractive Index · The Lorentz Oscillator Model · Complex Refractive Index and the Extinction Coefficient · Kramers–Kronig Relations.

What is the difference between χ, ε, and ε_r, and which one do I actually use?

Three names for one piece of physics. The susceptibility $\chi$ is the raw response coefficient, $\mathbf{P}=\varepsilon_0\chi\mathbf{E}$ (dimensionless). The permittivity $\varepsilon=\varepsilon_0(1+\chi)$ carries units of $\mathrm{F/m}$ and is what appears inside Maxwell's equations once you hide the bound charge. The relative permittivity $\varepsilon_r=\varepsilon/\varepsilon_0=1+\chi$ is the dimensionless "dielectric constant" a catalogue lists. For optics you almost always want $n=\sqrt{\varepsilon_r}=\sqrt{1+\chi}$ , so in practice you read $\varepsilon_r$ (or $n$ ) at the optical frequency and never touch $\varepsilon$ in SI units at all.

Why does the susceptibility have to be complex — isn't that just bookkeeping?

It encodes timing, which is real physics. With the convention $\mathbf{E}\propto e^{-i\omega t}$ , the phase of $\chi$ is the phase lag between the driven polarization and the field. A purely real $\chi$ means the charges move exactly in step with the field, storing and returning energy each cycle with no net loss — a transparent medium. A positive imaginary part $\chi''$ means the polarization lags, so over a cycle the field does net positive work on the charges: the medium absorbs. So $\chi''$ is not a mathematical convenience; it is the absorption coefficient in disguise, $\alpha\approx\omega\chi''/c$ .

Can the susceptibility be negative?

Yes — the real part $\chi'$ can be negative, and it routinely is just above a resonance (and throughout the X-ray region), where it can even push $\varepsilon_r=1+\chi'$ below one and give a refractive index less than unity. That means a phase velocity above $c$ , which is allowed because signals travel at the group velocity. What cannot happen in a passive medium is a negative imaginary part $\chi''<0$ : that would mean the medium adds energy to the wave (gain), which requires an external pump, as in a laser amplifier. Passive absorbers always have $\chi''\ge0$ .

How is this different from the electromagnetism article on susceptibility?

The electromagnetism owner, aq-em-linear-dielectrics-susceptibility-and-permittivity, works in electrostatics: constant fields, a real constant $\chi_e$ , and the bookkeeping of bound charge. That is where $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ and $\varepsilon_r=1+\chi_e$ are established. This article does one thing to that result: it lets the field oscillate at optical frequencies, so the constant becomes a complex function $\chi(\omega)$ . Everything optical — dispersion, absorption, the index — lives in that frequency dependence, which the static treatment cannot see. Same relation, promoted from a number to a function.

Where does the Lorentz oscillator model fit in?

The Lorentz model supplies the explicit shape of $\chi(\omega)$ that this article only sketches. Treating each bound electron as a damped, driven harmonic oscillator gives $\chi(\omega)\propto1/(\omega_0^2-\omega^2-i\gamma\omega)$ , a single resonance whose real part is the anomalous-dispersion wiggle and whose imaginary part is the absorption peak drawn in the punchline figure. We recall that result by name from aq-optics-lorentz-oscillator-model rather than re-deriving it, because the derivation is owned there; here we need only that $\chi(\omega)$ is complex, resonant, and causal.

Why does a bigger susceptibility mean a slower wave?

Because the induced polarization re-radiates. A larger $\chi$ means the bound charges respond more strongly to the passing field, producing a larger secondary wave that superposes on the original slightly out of step. Adding the two over many atomic layers shifts the crests back — the wave propagates more slowly, by exactly the factor $n=\sqrt{1+\chi}$ . So "how easily the charge moves" ( $\chi$ ) and "how much the light slows" ( $n$ ) are the same statement, related by a square root through the wave equation. A vacuum ( $\chi=0$ ) has nothing to re-radiate, so light travels at $c$ .