How is Lorenz-Lorentz relationship possible?

In summary: The refractive index ##n=\sqrt{2}i## implies an infinite phase-velocity ##c/n=\frac{c}{\sqrt{2}i}=i\sqrt{\frac{c^{2}}{2}}##, which you might have guessed from the LL expression ##c^{2}=\frac{1}{\varepsilon\mu}##, where ##\mu=1## is the vacuum permeability. So at very high densities, the electromagnetic fields cannot penetrate the medium.In summary, the Lorentz-Lorenz theory predicts a theoretical limit on the maximum value of electric polarizability, which results in a maximum value for the refractive index. As the density of a dielectric increases
  • #1
snorkack
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Its form is:
(n2-1)/(n2+2)=(4π/3)Nam
There is one simple problem with it. Rearrange the left side and you get:
(n2+2-3)/(n2+2)=(4π/3)Nam
1-(3/(n2+2))=(4π/3)Nam
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)Nam approaches and exceeds unity?
 
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  • #2
snorkack said:
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)Nam 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!
 
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  • #3
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?
 
  • #4
snorkack said:
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.
 
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1. How does the Lorenz-Lorentz relationship explain the relationship between refractive index and density?

The Lorenz-Lorentz relationship, also known as the Lorentz-Lorenz equation, is a mathematical formula that describes the relationship between refractive index and density of a material. It states that the refractive index of a material is directly proportional to its density, with a constant factor known as the Lorentz-Lorenz factor. This relationship is based on the assumption that the material is homogeneous and isotropic, meaning that its properties are the same in all directions.

2. What is the significance of the Lorentz-Lorenz factor in the relationship?

The Lorentz-Lorenz factor is a constant value that is used to convert between the refractive index and density of a material. It is derived from the theory of classical electromagnetism and takes into account the polarizability of the material, which is a measure of how easily its electrons can be displaced by an electric field. The higher the polarizability, the higher the Lorentz-Lorenz factor and the stronger the relationship between refractive index and density.

3. How is the Lorenz-Lorentz relationship used in practical applications?

The Lorenz-Lorentz relationship is used in a variety of practical applications, including optics, materials science, and geophysics. In optics, it is used to calculate the refractive index of a material, which is important for designing lenses and other optical components. In materials science, it is used to study the properties of different materials and to determine their composition. In geophysics, it is used to study the Earth's interior and to understand the properties of different layers of the planet.

4. What are the limitations of the Lorenz-Lorentz relationship?

The Lorenz-Lorentz relationship is based on several assumptions, such as the homogeneity and isotropy of the material, and it may not accurately describe the behavior of all materials. For example, it does not take into account the effects of temperature, pressure, or chemical composition on the refractive index. Additionally, it is only applicable to materials that are transparent or translucent to light.

5. Are there any alternative equations or theories that describe the relationship between refractive index and density?

Yes, there are several alternative equations and theories that describe the relationship between refractive index and density. One of the most well-known is the Cauchy equation, which is an empirical formula that has been shown to accurately describe the refractive index of many materials. Other theories, such as the Kramers-Kronig relations and the Sellmeier equation, take into account the frequency of light and the dielectric properties of a material to describe its refractive index. However, the Lorenz-Lorentz relationship remains a useful and widely used tool in understanding the relationship between refractive index and density.

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