# Correlation between the refractive index and the density

The $$v$$ propagation speed of light in a transparent medium is related to the $$c$$ speed of light in vacuum through the relationship,

$$\boxed{n=c/v}$$

The constant $$n$$ is a pure number called refraction index of the material. Since the speed of light in transparent material media is always less than $$c$$, $$n\geq 1$$.

Is there any correlation between the refractive index and the density of the air, or of a generic material?

I have seen this Relation between density and refractive index of medium but it's off-topic with what I'd also like to know about mathematical passages.

• – The Photon Jul 13 at 22:03
• @ThePhoton It's not easy for me to understand at all. I certainly approve of your kindness in providing me with the pdf. What exactly should I read? Could you adapt something simple so that I understand please? Thank you. – Sebastiano Jul 14 at 9:29

## 1 Answer

Yes, the index of refraction of air does depend on the density of the air, usually expressed in terms of the air pressure rather than the density.

This effect limits the accuracy of displacement measurements by interferometry, particularly when measuring the displacement of a moving object which is producing turbulence (air pressure variations) in the air around it.

The fractional content of water vapor and CO2 in the air also affect the index of refraction measurably.

From some brief web research, there are widely accepted fitting formulas for these effects from Edlen (1966) updated in 1994 by Birch and Downs; and by Ciddor (1996). A presentation from the Canadian National Research Council gives formulas based on Edlen, Birch, and Downs: Sadly, the individual terms (particularly $$x$$, $$\sigma$$, and $$f$$) are not fully explained, so you'll have to work out exactly what they mean or go back to the primary sources for an explanation.

The US's NIST provides an online calculator based on Ciddor, and some helpful instructions. I also found a page where you can download Python code for calculating the refractive index based on Ciddor.

I don't find any simple formula that gives just the sensitivity of index to pressure, but from the NIST page it seems that a difference in air pressure of approximately 0.4 kPa (standard air pressure being 101.325 kPa) produces an index of refraction change of about 1 ppm (this number likely slightly variable depending on wavelength, temperature, air composition, etc).

• I can understand your English without a translator :-). The shape is impeccable and the explanation very clear. What would you say about my question if you made some personal considerations? Could you write that $n=n(\mathrm{air})$ or something similar to better understand the concept? In the meantime, I voted in favour and since no one or only a few answer me, I accept your answer. – Sebastiano Jul 14 at 15:36
• @Sebastiano, I'm not sure what you're asking. As far as I know, the index of air under standard conditions, at visible wavelengths, is between 1.00027 and 1.00028. – The Photon Jul 14 at 15:42
• A general relationship which explains that $n$ depends on the air can be written according to your opinions, adding your precious informations? Thank you very much. – Sebastiano Jul 14 at 15:44
• @Sebastiano, Edited to include the formulas from the Canadian presentation. I'm not sure what all terms in the formulas mean, however. – The Photon Jul 14 at 15:49
• Thank you very much truly. I have accepted before. Grazie infinite :-) – Sebastiano Jul 14 at 15:51