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For hard X-rays the refraction index of glass becomes slightly $n<1$ which means that the phase velocity (not group velocity) of X-rays inside glass is larger than $c$ the group velocity speed value of light in the vacuum.

This is not a problem of course since phase velocity according SR can exceed the group velocity of light in a medium.

My problem is under what mechanism and physical explanation this happens for X-rays travelling through glass?

What I'm actually asking is how is it possible a medium more dense than air to have a smaller than one refraction index 0<n<1 when hit with x-rays? This would mean that glass at the x-rays band has relative permittivity and permeability product smaller than 1: $$ \begin{array}{l} \mathrm{r}=\frac{1}{\sqrt{\mu \varepsilon}}=\frac{1}{\sqrt{\mu_0 \mu_r \varepsilon_0 \varepsilon_{\mathrm{r}}}}=\frac{1}{\mu_{\mathrm{r}} \varepsilon_{\mathrm{r}}}=\frac{\mathrm{C}}{\sqrt{\mu_{\mathrm{r}} \varepsilon_{\mathrm{r}}}} \\ \therefore n=\frac{\mathrm{c}}{\mathrm{r}}=\sqrt{\mu_{\mathrm{r}} \varepsilon_{\mathrm{r}}} \end{array} $$

Is it because the capacitive reactance of glass $X_{C}$ becomes smaller as light frequency increases?

But I prefer if possible a quantum physical expatiation.

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  • $\begingroup$ As a starter reference this thread here, quora.com/Why-is-the-refractive-index-of-x-rays-less-than-1 $\endgroup$
    – Markoul11
    Commented Dec 25, 2023 at 20:54
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    $\begingroup$ this has a lot of answers. researchgate.net/post/… $\endgroup$
    – anna v
    Commented Dec 26, 2023 at 6:02
  • $\begingroup$ What I'm actually asking is how a medium more dense than air can have a smaller refraction index n<1 when hit with x-rays? This would mean that glass at the x-rays band has relative permittivity and permeability product smaller than 1: $$ \begin{array}{l} \mathrm{r}=\frac{1}{\sqrt{\mu \varepsilon}}=\frac{1}{\sqrt{\mu_0 \mu_r \varepsilon_0 \varepsilon_{\mathrm{r}}}}=\frac{1}{\mu_{\mathrm{r}} \varepsilon_{\mathrm{r}}}=\frac{\mathrm{C}}{\sqrt{\mu_{\mathrm{r}} \varepsilon_{\mathrm{r}}}} \\ \therefore \frac{\mathrm{c}}{\mathrm{r}}=\sqrt{\mu_{\mathrm{r}} \varepsilon_{\mathrm{r}}} \end{array} $$ $\endgroup$
    – Markoul11
    Commented Dec 26, 2023 at 11:34
  • $\begingroup$ Also another good question would be, has air also a rafraction index smaller than 1 for X-rays or is this property unique only for some solids like glass? $\endgroup$
    – Markoul11
    Commented Dec 26, 2023 at 11:50

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Phase velocity can be faster than light without causality paradoxes. A related example is to take two straight lines at an angle to each other. They cross at a point. Move one of the lines, and the point where they cross moves. If the angle is shallow, the point where they cross moves faster than the line. As the angle gets shallower, there is no limit to how fast the point can move. This is OK. Nothing physical moved very fast. Two different points on the string are now on top of each other.

Phase velocity is very much the same. Think of two plane waves at an angle to each other. The position of the crests of each wave is a series of parallel lines. The lines move in the direction the waves travel.

The total wave is the sum of the two waves. The point where the lines cross is where the maxima of the total wave is. The maximum can move at any speed. Nothing physical moves that fast. It is just the location where the crests of two waves cross. Different points on the two waves overlap at different times.


Addendum

I found a really great explaination for why xrays go faster than light in glass from 3blue1brown. It is the 4th in a series of videos on some curious optical effects.

How light can appear faster than c, why it bends, and other questions | Optics puzzles 4

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  • $\begingroup$ I agree this is a good example, $\endgroup$
    – anna v
    Commented Dec 26, 2023 at 6:04
  • $\begingroup$ While this is another example of a mathematical object moving faster than light, the actual question stays unanswered. What physical mechanism propagates wavefronts faster than causality? $\endgroup$
    – A. P.
    Commented Dec 26, 2023 at 10:02
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    $\begingroup$ I did not ask why phase velocity greater than c is not breaking causality but instead why x-rays in glass have such a high phase velocity? What is the physical explanation? Are the electrons wiggling with opposite phase to the incident light and this high speed is due to some interference effect? I mean with υ=λf and f value fixed this means that the wavelength of the x-rays inside the glass must get larger? Right? Also to keep thing simple, let's assume the incident angle of the x-rays to the z-axis is zero. $\endgroup$
    – Markoul11
    Commented Dec 26, 2023 at 11:01
  • $\begingroup$ Apologies. I missed that. $\endgroup$
    – mmesser314
    Commented Dec 26, 2023 at 15:07

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