How is Lorenz-Lorentz relationship possible?

[snorkack]

Its form is:
(n²-1)/(n²+2)=(4π/3)Na_m
There is one simple problem with it. Rearrange the left side and you get:
(n²+2-3)/(n²+2)=(4π/3)Na_m
1-(3/(n²+2))=(4π/3)Na_m
As you see, the left side cannot reach unity for arbitrarily large n².
But there is no reason why N cannot be arbitrarily large!
How does n behave at high densities of dielectrics, where (4π/3)Na_m approaches and exceeds unity?

 

[renormalize]

As you see, the left side cannot reach unity for arbitrarily large n².
But there is no reason why N cannot be arbitrarily large!
How does n behave at high densities of dielectrics, where (4π/3)Na_m approaches and exceeds unity?

This is not a limitation on the refractive index $n$ but rather a theoretical limit on the maximum value of the electric polarizability $\alpha$. If you solve the Lorentz-Lorenz (LL) equation for $\alpha$ and take the limit $n\rightarrow\infty$ you get:$$\alpha=\frac{3}{4\pi N}\left(\frac{n^{2}-1}{n^{2}+2}\right)\lt\frac{3}{4\pi N}\thickapprox\frac{0.24}{N}$$Now let's put in some numbers: for an ideal gas at $20^{\circ}\text{C},1\text{ atm}$ we have $N^{-1}\simeq4\times10^{-20}\text{cm}^{3}$, so LL predicts that $\alpha\lesssim10^{-20}\text{cm}^{3}$. And indeed, if you check tables of atomic and molecular polarizabilities, you discover that they all fall in the range $10^{-25}-10^{-21}\text{cm}^3$. Another successful prediction of Lorentz-Lorenz theory!

 

[snorkack]

But the problem is, since all substances are compressible, N has no upper bound and cannot have. So what does n do as N increases?

 

[renormalize]

So what does n do as N increases?

Well, the Lorentz-Lorenz relation is empirically most accurate for dilute gases, and reasonably good for dense gases and some liquids. But that said, we can just plow ahead and see what LL predicts for arbitrarily high concentrations. Let's start by substituting $n^{2}=\varepsilon$, where $\varepsilon$ is the relative permittivity (dielectric constant), into the LL relation to get the Clausius-Mossotti form $\alpha=\frac{3}{4\pi N}\left(\frac{\varepsilon-1}{\varepsilon+2}\right)$. Solving for the permittivity gives $\varepsilon=n^{2}=\frac{9}{3-4\pi N\alpha}-2$. In the ultra-dense limit $N\rightarrow\infty$, we see that the permittivity $\varepsilon=-2$ is negative, and that the refractive-index becomes purely imaginary, $n=\sqrt{2}i$. An electromagnetic plane-wave normally-incident from vacuum onto the surface of such an ultra-dense dielectric medium cannot propagate inside. Instead, the wave is perfectly reflected since the reflection coefficient $\frac{\sqrt{2}i-1}{\sqrt{2}i+1}$ has magnitude $1$. This LL prediction for an infinitely-dense medium seems physically reasonable.